I noticed that sometimes—not always, but often enough to be a problem—when I try to send the computer into hibernation it doesn't work. Instead of the hibernation screen, I get dumped back to the Desktop with a warning balloon popping up from the System Tray that reads: Insufficient system resources exist to complete the API.

Bewildered, I set out to find out what it might mean. As it turns out, Microsoft has documented this failure to hibernate as a known bug. It turns up under the following circumstances:

You're running Windows XP Service Pack 2, Windows XP Tablet PC Edition 2005, or Windows XP Media Center Edition 2005.
Your computer has more than 1GB of physical memory.
The system's memory is currently fragmented—meaning that applications have staked out enough physical memory that a large enough contiguous block of free memory to start the hibernation process can't be reserved. The more memory you have in the system to begin with, the bigger a block of free memory is needed.
Evidently the memory fragmentation issue is the key; many times, hibernation failed after I'd already run a number of other programs. Sometimes I was able to get hibernation to work again after logging off (i.e., after freeing up some of the memory in use), but that didn't always work either.

Microsoft has a fix for this, ostensibly set to go into the next Service Pack for Windows XP, but if you want to address it now you can call Product Support Services and ask for a free hotfix as referenced in the above article (909095). There's no charge for this call.

Note: Vista does not suffer from this issue.