so I asked a similar question to this with regard to handling a rejection/wait list and here's what people told me...
Straight up rejection --> LOI likely will be wasted time
Wait list --> Dr L responded to this and said ", if the program has a "wait list" and you send a note or email and have an attending call or write on your behalf that could put you potentially higher on the list (of course assuming that you're already a competitive applicant)." As such, in my mind if it is a place that you want to go, then it's a good idea. If you're just trying to get more interviews but you don't want to go there it might not be the best idea.
One of the other guys, who's a resident now i think said "One of my strategies to get interviews was if I did not receive a notice of rejection or acceptance from a program that was going to have an interview in a week or two..if I really was interested in the program, i would send an email to the program director, and residency coordinator expressing my interest, especially since I had heard that others were granted interviews. "
So, bottom line, if you really want to go to school X, and they have offered interviews and you are on the wait list/haven't been rejected, i probably won't hurt you. If it's your dream to go to school X, I would definitely do it and have an attending/program director/chairman call on your behalf.
I wouldn't, however, do it to programs that you randomly shotgun applied to and otherwise wouldnt' have been upset if they had straight up rejected you..cause they're gonna ask why you want that program so bad...
just my 2 cents
TCBO1