I read thru the AAMC website once about the deadline once and they had to put in a deadline to force people to pick a school. They consider it "unethical" to hold multiple positions after this date and as such AAMCAS "supposedly" has a list that they send out to each of the school for every student that is accepted and if a student is still holding more than one spot after this date, they have the right to take back their offer of acceptance and give it to another.
Unfortunately, this only applies to allopathic schools. Each osteopathic school has their own deadlines. Usually there is some kind of tuition deposit due date that is in addition to the acceptance deposit deadline. For many school it may be May 15th but you would need to check with each individual school to know for sure. TUCOM has a May 15th deadline.