I figured out the problem: The DreamPi and/or its USB dial-up modem had overheated and required time to cool down. I gave it a lot of time to cool, and a few hours later when I tried it, it all worked just fine!
I really ought to take my own advice and move them to a cooler place, but the place it is at is what works best and is most convenient with the arrangement of the TV stand and the consoles sitting atop it.
However, as I missed the scheduled session for The Next Tetris: On-line Edition, I have created a new scheduled game-play session for January 2nd. Please join me for Tetris at 9 P.M. Mountain Time
! The details are in the post I created over in Dreamcast Talk's section for The Next Tetris: Online Edition (which is linked to earlier in this paragraph).