I ended up waiting an hour before calling back to report that my digital cable is still broken. This time, the lady had me unplug the converter box and plug it back in. Still have the problem. She then booked an appointment to have someone come over tomorrow afternoon to look at it. Said that everyone had already gone home today and couldn't come out. OK. Fine. Whatever.
So I poked around on their website and found another number that pretended to be a 24-hour repair service number. I called it. Turns out it's just a number that you can call any time to report a problem, they don't actually do repair work 24 hours a day. So they're still on tap to come out tomorrow. I did ask the lady to credit my account since I'm not getting the digital cable service that I'm paying for and she at least agreed to do that.
One of the things I really enjoy doing on Sunday evenings is watching "The Wire" on HBO. Well, it looks like that's not going to happen tonight.