Need to eliminate HTTP Status 500 errors...

I have a web based java application running on multiple application servers.  All experience the same issue(s).  The heart of the problem is that while using the application, at any given time (fairly intermittent, no discernible pattern that we see) trying to get to no particular page, nor execute any specific functionality, we can get greeted with a page indicating:

HTTP Status 500 - Trying to process transaction <whatever we were doing / page we were trying to get to>
type: Status Report
message: Trying to process <fill in the action here>
description: The server encountered an internal error that prevented it from fulfilling this request.

Apache Tomcat 5.0.28  

A few very important points of note (I think if we solve #2- we solve the issue):
       1. Seems to be less frequent, but still happens, when there is less load (fewer users....less network traffic....however you want to view it).
       2. The application has a splash/home page that contains a window/box that is used for putting out announcements to the users.  When using IE as the browser and accessing the server/page using the server name - this box/window within the page displays this same HTTP 500 error ALL THE TIME without fail.   When using IE and accessing the server/page by IP, the announcement page displays correctly in the window ALL THE TIME without fail.   When using Firefox to access the server/page, this announcement window/box ALWAYS works regardless of using server name or IP.  (but, yes, we still do get the intermittent 500's using Firefox).
     3.  The application developers have this deployed at several other locations and indication is that none of them have ever experienced this, mean the 500's in the app or in the announcement box (and I do trust those telling me this).  They have reviewed the application, etc. and they feel it is something in the environment (network, DNS, etc.)
   4. Somewhat contradicting it being load/traffic related is the fact that the application screens are designed to continually refresh themselves (and they do) - meaning, if I were to leave a page up throughout the night it should still be up in the morning - but what we experience is that at some point during the evening (and there is next to no load and no users during the night) the refresh will come back with the 500 error.
  5. When we get the 500- we can usually just refresh the page (re-posting data or resending when need be) and more often than not, everything will be fine and the page will load- although we have had it occur where it happens multiple times in a row (but pretty infrequently).

We've tried all sort of things with DNS, network tweaking, etc. - and I am sure I'm forgetting some points of note above - but nothing we have tried has rid us of the problem.

Any help that can be offered is desperately appreciated...

Thanks so much...