Continuity mistake: In the beginning where Fletcher observes Andrew drumming for the first time, Fletcher hangs his jacket on a hook. A few seconds later after Andrew notices he's gone, there's no jacket hanging there. Shortly after that we hear the door open and it's reappeared. (00:03:20)
Continuity mistake: In the last scene, when the drummer starts the "caravan" song, the bass player on the left starts playing. But in the next shot he isn't.
Chosen answer: No, it is not normal to bleed while playing drums, as callouses develop on a drummer's hands through constant practising and playing. But, if a drummer plays hard enough and fast enough using the same drum sticks, friction between wood and skin can develop and rupture toughened skin. In the context of the film, the drummer is being pushed beyond his limits by the music conductor. Additionally, he is pushing himself to be better, so the bleeding is a consequence of that drive for excellence.
Scott215