I did some searching on the net and found some different articles that had a variety of suggestions for fixing the problem. These boiled down to:
- Turning off Messages and FaceTime in Settings
- Resetting Network Settings
But I was awfully close to doing just that when I tried using the Restore iPhone... option in iTunes (with the iPhone connected to my Mac.) That was when it told me that I needed to first turn off the Find My iPhone utility.
That was interesting, because other instructions had talked about turning off Messages and FaceTime, but hadn’t said anything about the Find My iPhone function.
I turned that setting off, made sure everything else (Messages and FaceTime) was off, then reset the Network Settings (from General in the Settings control panel on the iPhone, not using iTunes) and then I went and turned on Messages and voilà; it worked.
I’m pretty sure that’s all I did - though I had tried turning the iPhone off and on just before that, so it’s possible there was a longer sequence of actions needed to successfully do this - but if you’re having this problem, try turning off the Find My iPhone as well as iMessages and FaceTime to see if that fixes things.