It's hard to describe the "limits" of how some idiom can be used, because its use may be quite broad, with any number of variations and nuances. In general, to be "done with" something implies completion, but the details depend on context. For example, in FumbleFingers comment:
I'm done with John
implies the speaker does not want any more contact with John, but what this means depends on what we already know about their relationship. It could be that they are romantic partners, or they are coworkers, or even they are family members who have been fighting. For example
I'm done with my brother. I can't even talk to him about anything remotely political without him foaming at the mouth over everything he thinks is wrong with this country.
Note this may not imply that I never want to see my brother again, but rather that I'm exasperated with his behavior, and would prefer never to talk to him about a subject that leads to another tiresome tirade.
In a different context:
I'm done with my paper
naturally implies you've finished writing it, while
I'm done with her paper
means you have finished reading it, or possibly editing it. Along the same lines:
I'm done with exams
means you've completed your exams for the current school period. Meanwhile, Lambie's comment:
I'm done answering uninteresting questions
implies a certain degree of frustration with the quality of recently asked questions.
As I said, quite broad.