I do not have problems using Firefox ver 2 and using a dedicated IP address.
Here are some possibilities.
One reason is cookie settings [setting on the user end or the server end] Sorry - can't tell you exactly what to look for but I believe it would have something to do with A) allowing cookies or B) persistence, or how long the cookie is set to last.
Dial up accounts like AOL can give your computer a dynamic IP and if it changes then your session with the server gets messed up. Wireless networking can also do this if the signal is lost then comes back.
There are all kinds of programs and plugins that are supposed to make life easier but end up causing problems. Examples: background downloaders, and forum or blog 'helper' apps.
This can also happen when using a proxy server for internet access.
Firewalls and security software can cause this.
Power saving settings can cause this.
None of the things mentioned above specifically causes the problem - plenty of people use them without issues. Usually it is just some hard to find setting somewhere.
For troubleshooting, I would suggest installing a current version of an Internet Browser that has never been installed on your computer before (often previous settings just follow over if all you do is upgrade something you have used before). If with the default settings of this new browser you DO NOT have problems, then you know it is something in the browser you were using. Otherwise, well, good luck - too many unknowns.