So here's how I'm handling it:
-Trying to make sure all are in by the 15th (doesn't look like its going to happen, but maybe 3 will be in by then - the required number for almost all the programs I'm applying to)
-I have redundancies. I need 3-4, I have asked 5 people for letters, and I may ask a 6th in the next week or two. This way even if a doc doesn't manage to get it in, its OK, I have backups. Docs are busy. Its even harder if you are nowhere near where they are located
In terms of programs, based on what I've seen on websites, many will consider your app even if you don't have all the LORs in in time. Some won't though. Some will have a required amount (e.g. 3) by the time of interview, others will require another amount (e.g. 2) but expect you to get in another LOR by interview time or rank time, some will review your app regardless of what their website requirements say. Basically it varies a lot program to program.
I would try to get as many in as possible by Sept 15th, but not sweat it too much if they don't get in until Oct 1st, afterall the MSPEs don't go out until then anyways, and some programs won't review apps until then regardless.
