Assuming cell tower gremlins, I didn't think too much about it, but as a precaution returned home and restored and re-synced my phone.
Next day I made another call or two with no problems, then patted myself on the back for dealing with the problem on my own. Then the same symptoms returned. Not only were outgoing calls failing, but incoming calls failed in the same way.
A trip to the Apple Store was in order, so I made an appointment and sauntered on in. The tech was helpful. Tried the usual tricks like restarting the phone, resetting network settings etc, but nothing solved it. We switched SIMs with another phone, but no joy there. He suggested I call AT&T and have them 're-provision' the phone. Did that but no dice.