[fourstringfuror]

I tried a couple of things, one of which seemed to work. First, I did a hard reset. After rebooting, I deleted all the accounts, waited 10 minutes, then added back each one. Then, another reboot. Finally, I registered the device under Settings | About | Category | Network | Register BlackBerry.

So far, so good. I'll update the post later in case anyone else comes looking for a solution.