I submitted a support ticket, and the OBi staff were very helpful. The problem was with an iPhone app I had been using, GV Mobile +. I really liked that app, and it had worked fine for a long time (until this morning, as far as I can tell). But unless I'm missing something, something must've changed recently. Anyway, I uninstalled it from my phone, and also removed any GV Mobile + settings in my GV configuration. That did it.
The support staffer who helped recommended the Google Hangouts app. That I guess would work, but since I've really been trying to avoid joining Google+, I installed the original Google Voice app (also developed by Google), and that appears to work just fine too. It hasn't bee updated since September, so whether Google will continue supporting it or, finally, force me to join Google+, I don't know. But it works for now.