Well, it appears I've found my answer.
A problem has been discovered in the Firefox version of Java 1.6.0_03
If you know anyone who is having issues with a Java applet using 1.6.0_r3, please let them know about this.
Although only some servers are affected by this issue, I can confirm that all the affected servers I've tested WILL work with Internet Explorer and Opera using the same version of Java. Apparently, only Firefox is affected.
I will also test Safari and Firefox on my old Mac to see if the OSX version of Java has this issue as well.
I guess this will teach me to use different browsers when testing for Java applet compatibility, and not just HTML layout.