Is it possible to complete the last year of a German Diplom in another university if ineligible in original university? I have a friend who was studying civil engineering in Dresden, Germany. He somehow got a very... VERY bad case of electronic addiction (he would just play online games and only procrastinate) and he lost connection to reality. He also had to go to rehab for this.

He is doing a Diplom, he has to write his final dissertation. He has over 20 semesters in the degree, it was a bit of a start and stop. This is a German 'diplom' meaning it is equal to a masters' so he holds no previous degrees.  

Now he is ready to do the last part of his degree but he has been waiting for a decision from the university as to if he can continue his degree. 

My question is, if this decision from the university is a "no", is it possible for him to complete his education? I mean is it possible for him to enroll in another university only for the last year. I think it is not possible in Germany to do this (something to do with having done too many semesters already), but maybe in another country.
Your friend seems to be very lucky, as the the civil-engineering diplom programme in Dresden is one of the very few in Germany that did not switch to bachelor/master yet (source). So if he did not automatically fail his studies due to some rule, he is likely to be able continue almost where he left them. For details, you would really have to check the university’s guidelines.
If this is not possible for some reason, there are a few universities where the diplom programmes have not fully ended yet, but I would judge it unlikely that any of them allows for new students to enrol, even if they only need to write a thesis.

More generally, there is a general German law (Hochschulrahmengesetz § 15 (3)), that allows students to transfer accomplished achievements:

Zum Nachweis von Studien- und Prüfungsleistungen soll ein Leistungspunktsystem geschaffen werden, das auch die Übertragung erbrachter Leistungen auf andere Studiengänge derselben oder einer anderen Hochschule ermöglicht.
(For certifying studying or examination accomplishments, a point-based system system shall be created that allows to transfer accomplishments to other degree programmes of the same or another university.)