Q:

Can something be actually possible yet logically impossible?

By actual possibility I mean the possibility which is implied by ability or power.
By logical possibility I mean whether concepts of reality contradict each other or not.
I believe that knowledge and reason are accurate reflections of the nature of reality. Therefore I am not inclined to believe that something which is actually possible in reality (like a free choice) could also be logically impossible (for example, an allegedly true prediction about the outcome of the choice). If the possibilities indicated by reasoning do not match the actual possibilities of reality, then it seems to me there must be a false premise which would explain the discrepancy.
Can anyone prove this either way? Can the actually possible also be logically impossible?
Can something be within the limits of ability/power/opportunity yet also create a contradiction if it occurred?
It seems that merely asking this question presupposes that logic and knowledge do indeed reflect the nature of reality in an accurate manner.

A:

If we use your own definitions,

By actual possibility I mean the possibility which is implied by ability or power.
  By logical possibility I mean whether concepts of reality contradict each other or not.

then I would say that yes, it is straightforward to have something which is logically impossible but actually possible. The reason for this is that you do not specify which concepts — or models — of reality you are concerned with.
For a half-serious example, consider the dramatic revelation in Star Wars: Episode V (The Empire Strikes Back). Luke's reaction to the revelation that Darth Vader is his father is disbelief: that it is impossible. Whatever emotions drove this reaction, it is likely that this was his reaction to the fact that the assertion contradicted his mental model of Darth Vader and Anakin Skywalker being seperate individuals, one of whom literally killed the other, as Ben Kenobi described in Episode IV. Relative to this model of reality, it was logically impossible for Darth Vader to be Anakin Skywalker, because it violated an assumption used in constructing the model. But if we accept what is asserted or demonstrated by the later Star Wars films, it was in fact actually possible, and indeed true.
More seriously, discovering things which are actually possible but which are logically (more accurately, theoretically) impossible, is a good approximation to how science works according to Popper: by falsification. If a theory predicts that something should not happen (or is impossible to make happen), but which subsequently does happen, this invalidates the theory. For this reason, I would prefer to call this "theoretical", rather than "logical", impossibility, because it places the fault clearly where the failure of the model can more easily be remedied — by improving or replacing the theory.
In modern science, where we accept that probability may be an unavoidable feature of physical theories, we are presented with a more complicated situation. When a theory predicts one probability distribution, but experiment produces another, has the theory been falsified even if we assume that the experiment was "executed perfectly"? With probabilities, there is always of course a non-zero chance of freak occurrances in which events drawn from one distribution resemble another. This is of course less likely, the more random samples you take, but in most cases you cannot actually rule out the possibility that one distribution will in practise, with a finite number of samples, produce the curve of another. We then move from impossibility to improbability — where we might ask whether something is actually probable while being theoretically (or logically) improbable. This is a somewhat more nuanced, but still essentially Popperian notion of falsifiability: we accept that there can be such events, and that when they arise they indicate a failure of the theory.

A:

The short answer is no: if something's logically impossible, then it isn't going to happen, not no way, not no how. 
The distinction between logical possibility and other kinds is a little subtle though. It is possible to distinguish various kinds of possibility. 

x is logically possible if and only if it the existence of x entails no contradictions. 

The space of logical possibility is vast. There are dragons in logical space. There are worlds in which magic exists. There are worlds in which there are no physical laws. There are worlds in which nothing exists. The only things that there aren't in logical space are impossible things like round squares, or largest prime numbers. 

x is physically possible if and only if the existence of x entails no violations of the laws of physics, as we currently understand them. 

Everything that is physically possible is logically possible, but the converse isn't true. There is no magic in any physically possible world, because the existence of magic would violate the law of conservation of energy. 

x is technologically possible if and only if the existence of x is compatible with currently existing technology. 

Again, technological possibility is a subset of physical possibility. Faster than light transportation is not technologically possible (at the present) but perhaps it isn't physically possible. 
The important point is that what is physically possible (what is compatible with the laws of nature) or what is humanly (compatible with human abilities) or technologically (compatible with current technology) possible is always a subset of logical possibility. 
Take for instance, the question of God's omnipotence. God can do anything. "But God can't make round squares!" you say, "Therefore he can't really be omnipotent after all." 
Well no. It isn't a limitation of God's power not to be able to make a round square because there is simply nothing that a round square could be. There isn't something such that God fails to be able to create it. A round square is just an absurdity created by our power to combine words that look meaningful even if there isn't anything they could possibly refer to. 

