If you have a late interview at a rolling admission school, call them once a week, or even every few days, to ask whether there have been cancellations for earlier interview times.
My story: Mount Sinai originally invited me to interview in mid-March. When I called to confirm the date, I asked whether they had a "wait list" of some kind for earlier interview dates. They said "no", but I could call every once in a while to check for cancellations. I called a few days later, and a date 2 weeks earlier was available, but I declined it due to a conflict. I called a few days after that, and January 22nd was available!
So the point of my story is, don't disappointedly resign yourself to a late interview if you really love the school (Sinai is my top choice). Try other ways to get what you want. But above all, if you're planning on calling an admissions office multiple times, be very very nice to them. They have a difficult job. And, more often than not, they will remember talking to you!
