Big thanks to Larry Hannay who manifested the bug on his machine, provided me with the exceptions that the applet was throwing, and verified that the fix I made did indeed fix the bug.
Turns out it's a problem in IE's Java class library. When laying out your components with the AWT border layout, it's supposed to assume you're centering a component if you don't provide any constraints. Or. . .
should be equivalent to. . .
Some class libraries, though, throw an exception rather than assuming that the control should be centered. I assume it's something that was later fixed, as I never saw the problem even though we were running IE 5.0.
Thanks also to the folks who emailed me with your experiences and suggestions as to what could be the problem. I appreciate it.