For me, this was happening on a Lenovo T61 with 2GB of RAM… and for every time that I tried to hibernate.

Microsoft Knowledge Base Article (KB909095) : 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.

This problem typically occurs when the computer uses 1 gigabyte (GB) or more of RAM.

Hotfix for KB909095

Update 9:41PM: Hotfix worked great.