I've asked many residents this same question and have gotten pretty much the same answer...apply to many schools and then interview at about 8-12 places (avg being about 9-10 from those I talked to). They said they felt comfortable about matching with that many interviews and said taking more would not be helpful because of interview burnout. Unfortunately, we have to interview for preliminary programs as well...so this probably adds to the burnout factor...hope this helps some.
In regards to preliminary interviews, I have heard many different stories ranging from interviewing at one program to as many as ten.