IS this a Dell? Some Dell computers will go into some damned low-power mode, and the power light will change to Orange. The only way I have been able to fix it, was to put in a new power supply. I've done this for 2 workstations already.
Check your Power Light, is it Green or Orange, or blinking? If the computer is in sleep mode, it will Blink on and off once a second. Other blinking patterns are error codes. Sometimes a system will just get stuck in Sleep mode and you have to force it back on.
When you are trying to wake the system, does the Monitor come back on? I've had computers stall like this.
Try changing your advanced power settings:
Usually you can get to it by Control Panel, Power, Power Scheme, and open your scheme and go into Advanced. Try changing the values for "hard Drive Power Down", change it to 45 minutes. Also, try setting it so that the Monitor does not turn off, just enable toe screensaver after 15 minutes or so, then test it, make sure it recovers from Screensaver.
I don't know if you are just telling your monitor to go off or if you are allowing the computer to go into Sleep mode, I would change the Monitor Off to "Never" and just tell it to sleep after an hour of non-use.
Just try different variations, with the monitor turning off or staying on. You can always choose a Blank Screensaver.