Two reasons.

They give up or they ran out of tokens.

Personally, I've been in a position where it goes on for too long, and I just give up mainly because I rather not lose more tokens to a person who probably has no problem "reloading" their tokens. I've also experience where the other person takes it to the point until the other person runs out of tokens.

Sometimes, I feel like some of these players are bidding to "hoard" the players. They don't really need them, but they just want a full 4 or 5 star team.

My worse experience was yesterday. Going through 4 rounds (if I'm not mistaken), and the other person still has tokens. I gave up in the end because clearly that person has a lot more tokens than I do, and I was running out of tokens. It wouldn't make sense that I continue bidding, would it?

Sometimes, the bidding process makes me want to quit the game. You work hard to earn the tokens, but in the end, you don't win the bidding. There was a season where not a single player I bid for, I won.