Today, it stopped doing that, and instead went into this state where it would work for 15 seconds, glitch out for a minute or so, work for 15 seconds or so, and repeat.
Once I realized that it was a pattern that was going to keep going, I decided to call Speakeasy's customer support and see if they could track it down this time. So, in a moment of internet-upness, I looked up the number, and then called.
Much to my surprise, I did not end up in a hold queue. After I got through the short phone tree, I got about five seconds of music and a "Hello, how can I help you?"
So, I explained the problem, and the support representative made understanding-the-issue noises and did some tests on their end, and found that the signal-to-noise margins were painfully low (9db down, 2db up). Apparently this is most commonly a problem with in-the-house wiring, and so he started by asking me to replace the phone line from the wall to the DSL modem. I found a relatively new spare one and did that, and the modem immediately reconnected, and he re-ran the tests a couple of times, along with doing a speed test, while I loaded a few web pages. Conclusion: margins now at 36db down, 28db up (or something in that range), which apparently is approximately perfect.
And that was that.
(Also, somewhere in there I mentioned that the modem was pretty warm to the touch; he said, "Yeah, they run pretty warm, and that's normal; it's when it's too hot to keep your hand on that you should worry.")
Moral of the story: If the phone line to your DSL modem is old, kinked (or, as in this case, tied in a knot to keep it from slipping out because the clip is broken), or otherwise aged, and you're having DSL issues, replace the phone line. Also, good-quality customer support is worth paying for.