It really depends on the programs you are interested in attending. Though path overall is not as competitive as many specialties, certain programs are very competitive.
It also depends on how you look as an applicant. Good grades, high scores on USMLE, etc... then you don't need as many.
If you are interested in the big named places (ie. Stanford, Mass General, etc....) then put in a couple backups as well.
If you have bad stats, then make sure you have alot of backups.
Most of the applicants I have met went on around 8-10 interviews. This may have been overkill for some of them, but better to have a spot on match day than not just because you didn't want to go on many interviews.
I personally only went on 4 interviews. I knew I wanted to stay at my med school for residency and I knew they were planning on ranking me high. So the other places were just to see what else was around and have a couple backups just by chance they were not telling me the truth.