Why is my Mac "falling asleep" when it's told not to?
Is your computer actually sleeping, or is it a display sleep that looks like a real sleep?

What about the other options in Energy Saver, like "Put hard disk to sleep when possible" (unchecking this may help with being logged out of some chat programs) and "Wake for network access" (checking this helps with other chat programs)?

Also check the schedule button to make sure no sleep schedule got accidentally set.

In my experience, all of the sleep options are controlled from the Energy Saver panel, so I'd check your system preferences first.  If that doesn't do it, check the settings in any third-party system extensions you have that might deal with power or computer performance.