The computer will hang. This will happen more frequently when the system has been
on for a while or the room temperature is above normal.. If the unit is immediately
reset, the problem will quickly recur. If allowed to remain off and to
literally cool off, then it will again run normally for a while.
If the fan in the power supply is not working, you can tell by just putting your hand near the grill. If you don' t feel a breeze, then the fan is not working.
Sometimes blowing the fan out with a can of compressed air will help. Be prepared for a big cloud of dust!
The CPU fan is tougher to check; you have to open the case and look to see if it is still working. CPU fans in tower systems are prone to failure. Starting/operating against gravity seems to wear them out faster than those in desktop systems.