Is it possible to be part of more than one team in the 2v2 ladder? If so, how many teams am I allowed to be in?

You're only allowed to be active (playing ladder games) on one team at a time.

You can make as many teams as you want, though, if you're especially fickle with staying with a teammate. Just note that it takes 10 games to be rated, and it's usually better to stick with one team.

It makes sense if you think about it: the game assigner could put you up against yourself.