user157: Does any Buddhist school or tradition make the assertion that all human beings have the capacity to be enlightened in this lifetime? Or to put it the other way are there groups of people who it just is not possible for them to become enlightened and they have to wait for a better rebirth? Do different traditions differ on this? Does any tradition make the claim that particular human beings will never be enlightened no matter what even in subsequent rebirths?

user-1: Anyone who simply changes the way they think could potentially be enlightened in this lifetime. The critical point is that people are not willing or able to change their viewpoint. If one thinks that what s/he is experiencing is happiness then that individual would not have use for a Dhamma expounded by a Buddha for the purpose of elimination of suffering. People only care about doctors when they get sick. If they don't even know that they are sick in the first place why would they go to a doctor much less take the medicine they prescribe?
In my opinion one of the biggest barriers to entry aside from obvious ones (like having a debilitating mental disability) are the three distortions of perception (sanna-vipallasa). Which three? Perceiving that which is inconstant as constant, perceiving that which is unsatisfactory as satisfactory, and perceiving that which is not-self as self. There are so many extremely intelligent individuals in my life that would be able to appreciate the Dhamma if only they were willing to consider the possibility that life is unsatisfactory. An individual can abandon wrong views if they're humble enough to admit to themselves that they could be wrong, otherwise they are stuck indefinitely in the perceptual cages that they've built for themselves.

user106: As far as I understood, the (Mahayana-) Mahaparinirvana-Sutra was composed some generations after the Buddha's death; but it might be interesting that it contains the new, previously unknown, notion of "icchantica(s)" - people who are not able to become enlightended because they are so ... evil? stupid? compassionless? (Don't know the exact idea).   

Update: Ah, here is some paraphrase from the presenting site, Dr. Tony Page:      


(...)
  i) That the noble Mahaparinirvana Sutra possesses the power to
  bring about "benefit, happiness and kindness for all beings" - with
  the one possible exception of those termed icchantikas, the most
  spiritually deluded of persons, who disparage the sutra and reject its
  teachings on the Tathagatagarbha; (...)

And here some excerpts from the Sutra (.pdf-file) itself:          

Let us hope always to learn Dharma. We shall draw out the tongues of
  those who slander the Buddha’s Wonderful Dharma.  (pg. 5)
The icchantika possesses the Buddha-Nature, but overspread by
  innumerable defilements, he cannot hope to get out [of his cocoon of
  defilements], analogous to the silkworm.  For this reason, he cannot
  gain the all-wonderful cause of Enlightenment, but repeats birth and
  death unendingly. (p.128)
All this comes about through the working of the divine medicine of
  the Mahayana Great Nirvana Sutra. All beings are deposited in peace,
  except the mahanaga and the icchantika (p.130)
Ever since, I have never once had occasion to commit myself to the
  evil actions of defilement and evil karmic relations, or to slander
  Wonderful Dharma, to become an iccantika, to be born with imperfect
  genital organs, or no genital organs, or dual genital organs, to act
  against my parents, to kill an arhat, to break a stupa or the law of
  the Sangha, to cause blood to flow from a Buddha’s body, or to commit
  the four grace offences.   (p.156)

Note, that this sutra (at least in its translation) refers in its text to itself , similar to a platform of a political party... 

Additional remark: for me things like those (the appearance of new ideas later in history, and for instance the idea of attacking people because they disparage the sutra and reject its teachings) are strong signals/markers/alarms that something is beginning to go wrong with the transmission of the dharma (I didn't see something similar in the palicanon-suttas) So I don't think, the idea of enlightnenment-disabled people is not historic in the palicanon.

user111: 
In the Ugghaṭitaññū Sutta (AN. ii. 135) the Buddha says that there are
  four kinds of person found in this world: those who are quick in
  acquiring, those who learn by means of a detailed exposition, those
  who may be guided, and those for whom the letter [of the Teaching]
  alone is the highest thing. In the Abhidhamma Piṭaka these are defined
  as follows:
What sort of person is quick in acquiring (ugghaṭitaññū)? The person
  for whom there is penetration of the Dhamma at the very time when it
  is being taught is called “quick in acquiring.”
What sort of person is one who learns by means of a detailed
  exposition (vipañcitaññū)? The person for whom there is penetration of
  the Dhamma when the meaning of what has been taught in brief is later
  analysed in detail is called “one who learns by means of a detailed
  exposition.”
What sort of person is one who may be guided (neyya)? The person for
  whom penetration of the Dhamma comes gradually by means of recitation,
  questioning, proper attention, and by serving, cultivating and waiting
  upon kalyānamittas is called “one who may be guided.”
What sort of person is one for whom the letter alone is the highest
  thing (padaparama)? The person for whom penetration of the Dhamma will
  not come in this life, however much [of the Teaching] he may hear and
  speak and bear in mind or recite, is called to be “one for whom the
  letter alone is the highest thing.” (Puggalapaññatti 41-2; Designation
  of Human Types 58)
Regarding the fourth type, the padaparama, his lack of potential for
  awakening in the present life may be due to a number of factors. Most
  padaparamas were already such before they were even born, when still
  in their mothers' wombs. Only humans conceived with a triple-rooted
  relinking consciousness (i.e., one accompanied by all three kusala
  roots: non-greed, non-hate and non-delusion) have the possibility of
  attaining jhāna or the noble path in their present life. Those with
  fewer kusala roots than this are all padaparamas.
Even those who have been reborn with such a consciousness may still be
  padaparamas. In the Puggalapaññati Atthakathā (PuggA. 184-5)
  Buddhaghosa lists six causes for incapacity to attain the paths and
  fruits in the present life; in each case the state is reckoned as one
  that lacks the requisite decisive support condition
  (upanissaya-paccaya). A similar list is also given in the
  Paṭisambhidāmagga (Paṭi. i. 123), but here I will use Buddhaghosa's as
  the Paṭisambhidāmagga's version does not supply any explanation.

Obstruction by kamma (kammāvaraṇa); meaning those who have committed one of the five anantariyaka kammas.
Obstruction by defilement (kilesāvaraṇa); meaning those who hold to any of the ten niyata wrong views ("there is no giving, no
  sacrifice... etc.").
Obstruction by kammic ripening (vipākāvaraṇa); meaning those who were reborn with only a double-rooted or a rootless relinking
  consciousness.
Lack of faith (assaddha); "one lacking faith in the Buddha, Dhamma and Saṅgha."
Lack of zeal/desire-to-act (acchandika); defined as being one who either lacks chanda in the sense of desiring to undertake what is
  kusala.
Being weak in wisdom (duppañña); defined as those in whom the bhavaṅga-citta of that lifetime lacks the mental factor of paññā (this
  in fact overlaps with #3, for it is the relinking consciousness that
  determines the character of the bhavaṅga-citta in any lifetime).
  Blockquote


Above is copied from the answer given here by ven. Dhammanado 

Does any tradition make the claim that particular human beings will
  never be enlightened no matter what even in subsequent rebirths?

It is said that those who are with Niyata Micchāditthi will be stuck in Samsara indefinitely. They are also called as beings who are tied to a pole in Samsara. They are said to be born in the Lokantarika hell when the world system gets destroyed. It is considered even worse than committing one of the 5 heinous crimes.

157: <p>Does any Buddhist school or tradition make the assertion that all human beings have the capacity to be enlightened in this lifetime? Or to put it the other way are there groups of people who it just is not possible for them to become enlightened and they have to wait for a better rebirth? Do different traditions differ on this? Does any tradition make the claim that particular human beings will never be enlightened no matter what even in subsequent rebirths?</p>
 -1: <p>Anyone who simply changes the way they think could potentially be enlightened in this lifetime. The critical point is that people are not willing or able to change their viewpoint. If one thinks that what s/he is experiencing is happiness then that individual would not have use for a Dhamma expounded by a Buddha for the purpose of elimination of suffering. People only care about doctors when they get sick. If they don't even know that they are sick in the first place why would they go to a doctor much less take the medicine they prescribe?</p>

<p>In my opinion one of the biggest barriers to entry aside from obvious ones (like having a debilitating mental disability) are the three distortions of perception (sanna-vipallasa). Which three? Perceiving that which is inconstant as constant, perceiving that which is unsatisfactory as satisfactory, and perceiving that which is not-self as self. There are so many extremely intelligent individuals in my life that would be able to appreciate the Dhamma if only they were willing to consider the possibility that life is unsatisfactory. An individual can abandon wrong views if they're humble enough to admit to themselves that they could be wrong, otherwise they are stuck indefinitely in the perceptual cages that they've built for themselves.</p>
 106: <p>As far as I understood, the (Mahayana-) Mahaparinirvana-Sutra was composed some generations after the Buddha's death; but it might be interesting that it contains the new, previously unknown, notion of <em>"icchantica(s)"</em> - people who are not able to become enlightended because they are so ... evil? stupid? compassionless? (Don't know the exact idea).   </p>

<p><hr>
<em>Update</em>: Ah, here is some paraphrase from the <a href="http://www.nirvanasutra.net/" rel="nofollow">presenting site, Dr. Tony Page</a>:      </p>

<blockquote>
  <p>(...)<br>
  i) That the noble Mahaparinirvana Sutra possesses the power to
  bring about "benefit, happiness and kindness for all beings" - with
  the one possible exception of those termed icchantikas, the most
  spiritually deluded of persons, who disparage the sutra and reject its
  teachings on the Tathagatagarbha; (...)</p>
</blockquote>

<p>And here some excerpts from the Sutra (<a href="http://www.nirvanasutra.net/convenient/Mahaparinirvana_Sutra_Yamamoto_Page_2007.pdf" rel="nofollow">.pdf-file</a>) itself:          </p>

<blockquote>
  <p>Let us hope always to learn Dharma. We shall draw out the tongues of
  those who slander the Buddha’s Wonderful Dharma.  (pg. 5)</p>
  
  <p>The icchantika possesses the Buddha-Nature, but overspread by
  innumerable defilements, he cannot hope to get out [of his cocoon of
  defilements], analogous to the silkworm.  For this reason, he cannot
  gain the all-wonderful cause of Enlightenment, but repeats birth and
  death unendingly. (p.128)</p>
  
  <p>All this comes about through the working of the divine medicine of
  the Mahayana Great Nirvana Sutra. All beings are deposited in peace,
  except the mahanaga and the icchantika (p.130)</p>
  
  <p>Ever since, I have never once had occasion to commit myself to the
  evil actions of defilement and evil karmic relations, or to slander
  Wonderful Dharma, to become an iccantika, to be born with imperfect
  genital organs, or no genital organs, or dual genital organs, to act
  against my parents, to kill an arhat, to break a stupa or the law of
  the Sangha, to cause blood to flow from a Buddha’s body, or to commit
  the four grace offences.   (p.156)</p>
</blockquote>

<p><em>Note, that this sutra (at least in its translation) refers in its text to <strong>itself</strong> , similar to a platform of a political party...</em>          </p>

<hr>

<p>Additional remark: for me things like those (the appearance of new ideas later in history, and for instance the idea of attacking people because they disparage <strong><em>the sutra</em></strong> and reject <strong><em>its</em></strong> teachings) are strong signals/markers/alarms that something is beginning to go wrong with the transmission of the dharma (I didn't see something similar in the palicanon-suttas) So I don't think, the idea of enlightnenment-disabled people is not historic in the palicanon.</p>
 111: <blockquote>
  <p>In the <strong>Ugghaṭitaññū Sutta (AN. ii. 135)</strong> the Buddha says that there are
  four kinds of person found in this world: those who are quick in
  acquiring, those who learn by means of a detailed exposition, those
  who may be guided, and those for whom the letter [of the Teaching]
  alone is the highest thing. In the Abhidhamma Piṭaka these are defined
  as follows:</p>
  
  <p>What sort of person is quick in acquiring (<strong>ugghaṭitaññū</strong>)? The person
  for whom there is penetration of the Dhamma at the very time when it
  is being taught is called “quick in acquiring.”</p>
  
  <p>What sort of person is one who learns by means of a detailed
  exposition (<strong>vipañcitaññū</strong>)? The person for whom there is penetration of
  the Dhamma when the meaning of what has been taught in brief is later
  analysed in detail is called “one who learns by means of a detailed
  exposition.”</p>
  
  <p>What sort of person is one who may be guided (<strong>neyya</strong>)? The person for
  whom penetration of the Dhamma comes gradually by means of recitation,
  questioning, proper attention, and by serving, cultivating and waiting
  upon kalyānamittas is called “one who may be guided.”</p>
  
  <p>What sort of person is one for whom the letter alone is the highest
  thing (<strong>padaparama</strong>)? The person for whom penetration of the Dhamma will
  not come in this life, however much [of the Teaching] he may hear and
  speak and bear in mind or recite, is called to be “one for whom the
  letter alone is the highest thing.” (Puggalapaññatti 41-2; Designation
  of Human Types 58)</p>
  
  <p>Regarding the fourth type, the padaparama, his lack of potential for
  awakening in the present life may be due to a number of factors. Most
  padaparamas were already such before they were even born, when still
  in their mothers' wombs. Only humans conceived with a triple-rooted
  relinking consciousness (i.e., one accompanied by all three kusala
  roots: non-greed, non-hate and non-delusion) have the possibility of
  attaining jhāna or the noble path in their present life. Those with
  fewer kusala roots than this are all padaparamas.</p>
  
  <p>Even those who have been reborn with such a consciousness may still be
  padaparamas. In the Puggalapaññati Atthakathā (PuggA. 184-5)
  Buddhaghosa lists six causes for incapacity to attain the paths and
  fruits in the present life; in each case the state is reckoned as one
  that lacks the requisite decisive support condition
  (upanissaya-paccaya). A similar list is also given in the
  Paṭisambhidāmagga (Paṭi. i. 123), but here I will use Buddhaghosa's as
  the Paṭisambhidāmagga's version does not supply any explanation.</p>
  
  <ol>
  <li>Obstruction by kamma (kammāvaraṇa); meaning those who have committed one of the <a href="https://en.wikipedia.org/wiki/Anantarika-karma"><strong>five anantariyaka kammas</strong></a>.</li>
  <li>Obstruction by defilement (kilesāvaraṇa); meaning those who hold to any of the ten niyata wrong views ("there is no giving, no
  sacrifice... etc.").</li>
  <li>Obstruction by kammic ripening (vipākāvaraṇa); meaning those who were reborn with only a double-rooted or a rootless relinking
  consciousness.</li>
  <li>Lack of faith (assaddha); "one lacking faith in the Buddha, Dhamma and Saṅgha."</li>
  <li>Lack of zeal/desire-to-act (acchandika); defined as being one who either lacks chanda in the sense of desiring to undertake what is
  kusala.</li>
  <li>Being weak in wisdom (duppañña); defined as those in whom the bhavaṅga-citta of that lifetime lacks the mental factor of paññā (this
  in fact overlaps with #3, for it is the relinking consciousness that
  determines the character of the bhavaṅga-citta in any lifetime).
  Blockquote</li>
  </ol>
</blockquote>

<p>Above is copied from the answer given <a href="http://www.dhammawheel.com/viewtopic.php?f=16&amp;t=1067&amp;start=20">here</a> by ven. Dhammanado </p>

<blockquote>
  <p>Does any tradition make the claim that particular human beings will
  never be enlightened no matter what even in subsequent rebirths?</p>
</blockquote>

<p>It is said that those who are with <a href="http://www.wisdomlib.org/definition/niyata-micch%C4%81ditthi/index.html">Niyata Micchāditthi</a> will be stuck in Samsara indefinitely. They are also called as beings who are tied to a pole in Samsara. They are said to be born in the Lokantarika hell when the world system gets destroyed. It is considered even worse than committing one of the 5 heinous crimes.</p>
