I don't think you would. Last year I submitted my AACOMAS in mid July. I received secondaries in early August and interviewed on the 2nd day of interviews at TUCOM, and interviewed at AZCOM in mid September and COMP at mid October.
The key is to get your secondaries and LORs out right when you get the secondary. In this way, you can probably afford to wait till even late June to send out your primary app while waiting for waitlists on your other schools.