I like to try and disprove results I am trying to understand.
Sometimes I'll give it a good month- I start by constructing perverse sequences and little heuristic arguments- when I'm onto something good I'll pursue it 'til it's rigorous: more often than not it turns out to be precisely the case disallowed by the theorem's assumptions.
After a while I start to paint my counter-arguments in generalities "An A is not a B if A has perversity X" and I look for the exact line in the proof where 'perversity X' is (perhaps implicitly) dealt with and debunked. Sooner or later I start to see what every line in the proof is for, moreover I start to see why it must be true.
It may not be the fastest way to learn, but it's certainly entertaining if you're having trouble sleeping!
One can also apply the same trick to definitions and constructions by trying to prove that they are useless/degenerate.