Bears have a few problems:

1. They can wear saddles before they're old enough to ride
2. They jump incredibly high, and if they come down at the same height they jumped from, you (not the bear) take fall damage
3. When quickly mounting and dismounting one, the game has a chance of attaching you to the bear, but all the controls are for the bear AI. It wanders around aimlessly and the only way to dismount from it is by quitting the world. It seems to happen more often if you hold shift while mounting it (I was trying to put a chest on, which also doesn't work).

That last one might be a problem for other mounts, too, but I've only noticed it in black bears.