The old SDN wisdom used to be 6 or 7 put you in a pretty good position, tho never really "safe."
Not sure that's true anymore. I went through the 07-08 cycle and I don't remember horror stories of people with multiple interviews not getting in anywhere. (I had 9 interviews, withdrew from one, and got 6 acceptances.)
But I browsed last cycle, and it seems that that was happening -- good stats, several interviews, no acceptances. Fair to say, it gets worse each year.
Good luck to all, except meltcher, or whatever.
Last year this topic was dominated by one prominent SDNer with stellar stats who, until really late in the cycle, had no acceptances, and he started multiple threads and it could have looked like an epidemic of high stats people not getting in, but he ended up with at least 2 - Pitt and WashU as I recall - and he is at the latter now. SDN name is WashMe - look him up, a decent chap. However, even he admitted that he had not done a good job applying - he either failed to apply to some of his state schools, and/or he applied late to them or other schools, thus his app list was very top heavy. There is almost always an explanation that help to explain these tales of woe...
Otherwise, I don't think it is correct to say that the goal line has really moved in recent cycles and that it is happening with more frequency - I agree with all posters who say no one is truly safe until they get an acceptance - but I do think that some number of interviews provides as much safety in this process as one can get, and it appears to fall in that 6 to 9 range, and while there are always exceptions, I don't recall anyone with that many interviews last year getting completely shut out, but there were a couple of people with just 3 or 4 interviews who didn't get in anywhere.
Like I said, I have 6, and I don't feel safe, but I do feel very hopeful.
Final note: IIRC the majority of matriculants only get into
one school, so it is a very thin line between success and failure for the majority of matriculants. And something like 60 percent of applicants get ZERO acceptances. Therefore, it shouldn't be hard to believe that that some highly qualified high stats applicants with a bunch of interviews occasionally fall through the cracks...but if you dig, there is usually a plausible explanation.