My application was submitted june 24th of last year, give or take a day, and didnt take my new DAT until after the summer. I got 2 pre-dec interviews and one acceptance. check predents.com for info (important dates) on accepted students and applicants. I think august is when apps start arriving at schools in large quantities, a june or july submission should keep you "ahead" of the game, some schools do not begin to review applications until july the earliest, some not until august.
Good luck to all those applyiing