Yes, by local phone I mean the phone that's plugged into the Obi box.
I figured out why it's doing it, but I don't know why it's doing it. I had three devices plugged into the box. A telephone, an alarm dialer that used the through output of the telephone, and a dial-in automation controller. The phone/dialer was on one port, the automation controller on the other.
Yesterday, doing some network rack cleanup, I decided the alarm dialer was no longer serving it's purpose and removed it from the system. Today, the phone connected to that port rings as normal. As the alarm dialer is incapable of picking up the line unless it rings something like 20 times, I have to assume that it was somehow suppressing the phone's ability to ring. Why this is happening, I have no idea.
Problem solved, but who knows why it did what it did.