Rebooting the switch cleared up all the problems. Since the switch and cameras had been happily coexisting in the same building for over two years I wondered what could have gone wrong all of a sudden. Then I remembered that about three months earlier I had installed a UPS to keep that switch and the connected POE cameras up during one of our all too frequent power failures. Apparently that switch needs a reboot every now and then, power failure induced or otherwise, in order to stay happy.
It must be running Windows.

Anyway, just one more thing to remember to check when the system goes south.