Late last year i was using my Dell laptop for a Team Foundation Server and having issues with it not going into hibernation...In my case it turned out to be the ATI video driver i was using.
Uninstalling this driver fixed my problem but there other folks having issues with computers with 2gig+ of RAM on Windows Server 2003. As described in the article: The computer occasionally does not hibernate and you receive an "Insufficient System Resources Exist to Complete the API" error message in Windows XP with Service Pack 2, in Windows XP Tablet PC Edition 2005, or in Windows XP Media Center Edition 2005
Looking at this article you will see it excludes Windows Server 2003 as one of the operating systems it applies to, but if you look at the Windows Server SP2 (Release Candidate) fix list you will see it is included!
List of updates in Windows Server 2003 Service Pack 2 Release Candidate
The download for Windows Server SP2 can be found at:
Windows Server 2003 Service Pack 2 Release Candidate
Even with this fix applied you may still run into this issue under certain scenarios. See the article below:
You cannot put a computer that has more than 4 GB of memory into hibernation in Windows XP and in Windows Server 2003