Today we played a 3 player game and one of us was very close to losing his last province; is this thing possible? I don't see why it wouldn't be possible. But wouldn't that "eliminate" that player from the game, making it hard for him to recover?
There is no rule preventing a player from losing his last province. It never happened in play testing because, usually, each player has at least one army in a capital of a province they control meaning there are easier provinces to go after.