Dell XPS 400 seems to have power issue - boots up sometimes and randomly powers off

This sounds like a classic CPU overheating issue. Can you check to make sure the CPU cooling fan is running properly.
Check for dust bunnies in the case and especially on the heatsink for the CPU.
Make sure the heatsink is properly attached and not loose on the CPU.
Also you may need to remove the heatsink and clean the thermal paste off and apply new thermal paste.
Here is the service manual for this model to help you in your endeavor.
http://support.dell.com/support/edocs/systems/xps400/sm/index.htm

This is not a known issue.

Is your system under warranty, if yes you should call Dell and get the fan and Heat sink assembly replaced. They will do troubleshooting on your system before dispatching Heat Sink and fan assembly for your PC. You should also refer to manual provided by sparkmaker to replace the Heatsink and fan assembly yourself.

In all probability it is the Heat sink and Fan assembly.

Before jumping to the CPU, if you have a extra power supply laying around give that a try. If youre having a loss in voltage in the power supply the Pc may power down randomly. You say most of the time that it powers back up, does it sometimes not power up then? If you have a warranty, go that route, if not try a power supply first before you go ripping your PC apart.

You could test the power supply.   i just bought one of these,
http://www.newegg.com/Product/Product.aspx?Item=N82E16899705003
I should have owned one years ago,
 
homemadebyX