Thanks, everyone, for your help. It appears that the situation had been resolved by the time I went back to reconnect the switches on the offending floor. Perhaps the culprit heard through the grapevine that it was a resident that caused the problem, and removed their router from the network.

But sure enough, last night the exact same thing happened again. I went in this morning and this time it was actually a different floor causing the same problem! Dumb kids! I started by narrowing down the floor, then the switch on that floor, then fortunately the cables were labeled for once so I could narrow it down to the room.

With the "arp -a" command (thanks, Bitt!), I was able to get the Mac. I knocked on the door, a guy opened up. I asked "do you have a router set up in here?" "Yeah, but it's not working! I moved it back to this other room here!" Sure enough, I walk back there and the cable from the wall is plugged into a LAN port. And there on the bottom was the matching MAC. Stupid kids!

Thanks again, folks. Hopefully this won't be a problem in the new building. If they'd agreed to the proposal I'd made a year ago for putting wireless in the building, we wouldn't have had this problem...