This year's tourney was 12 rounds, which should accommodate 4,096 teams.  We're currently at 3,807 US teams.  So, I think you've got it backwards.  If fewer than 290 more teams sign up, everyone should be able to play in the tourney.  If more than that sign up, presumably a few will get left out.  (At what point, if ever, does it go to 13 rounds?)