?? These are the most important references. Many excellent candidates won't have a back connection but when they are possible they can make an important difference two ways: One is the way described here (or, "this person was OK but doesn't sound like they would work out for the kind of thing you want -- that was where they failed with us too."). The other is the opposite, which has happened to me twice: "Oh that person? Look, if you don't take them let them know we'd really like them back" (on a candidate who barely made it through the interview but there was enough something there that we decided to do some background checking. Turned out he was simply nervous in front of people he didn't know and gave a terrible interview.). Without the background check we might have missed an excellent candidate with poor job-seeking skills.
Whenever I get a resume with a school on it that I or one of my colleagues attended I always check. I also once had an amazing candidate with 20 years experience in the lab doing just what we wanted. Their resume said that they had a degree from my schoo in X. I of course looked them up on the alum site and saw they had a degree in Y. Now both X and Y were in the set {Chemistry, Physics, Material Science, Chemical Engineering} so clearly had they simply told the truth it wouldn't have mattered -- the 20 years of experience would have said enough for anyone who actually cared about degrees after all that time. But since they lied...the resume went into the trash and the candidate was never brought in for an interview.