I found a Dell computer monitor from about 1999, with a Toshiba CRT in it. When I first power it up, the convergence on the bottom is TERRIBLE, but then the colors slowly converge themselves, and in about 15-20 seconds, the picture is fine.

Is there a problem here, or is that how it's designed? I've never seen a monitor do this before, so I'm not sure.

It almost seems to be a weak crt but that is just a guess, are you able to test that crt?

No way to test the CRT.

The CRT, while a tad dim, brightened up nicely with a slight tweak to the screen voltage, and I have never seen a weak CRT being the cause of poor convergence.

Try moving around the wires which connect the yoke to the board. There may be some sort of cross-interference.

Also, degauss it if it has that option at the press of a button, and see if it affects the problem.

Degaussing doesn't affect it. The monitor's picture is stable once it's been on for 20 seconds or so.

Also, the CRT reaches full brightness before the colors converge. The CRT is at full brightness after maybe 10-12 seconds, and the colors finish converging a few seconds later.

If it's fine after 20-30 seconds, I wouldn't worry about it. It might be normal for that monitor.