Is it normal for a rabbit to bite you if you put your hand in its cage?

Yes it is normal cuz the rabbit might be protecting its territory.

Take a look at the circumstances. It depends on how friendly the rabbit is normally. If this was unusual, then you may have been especially noisy and/or too fast in placing your hand inside the cage. Also if there are babies, the mother will defend them.

We did have a "crabby" rabbit that did not want anything to do with us, and she was always that way, even though she was never mistreated. We did get her at three months from a pet store, so she may not have been properly socialized.