Matthew, What you are referring to is Sleep mode and after a certain amount of time this sleep goes into hibernation mode.
For it to work properly you need the latest BIOS update for your motherboad, check with Compaq just to be sure you have the latest. The feature in the BIOS you need is called ACPI.
-also, and this may well be YOUR issue, -
In control panel - device manager check all the Human Interface Devices listed, and all the keyboards and mice ... right-click on each one and any that have a Power Management tab, make sure they are NOT checked to allow the computer to turn off this device to save power and make sure it IS checked to allow this device to wake the computer. Try these two things - they may solve your problem.
*** I'm not sure if disabling hybrid sleep will help but even if you do, you should also check the above items
If the problem persists,
In BIOS there may be an ACPI setting you can change. I have seen settings of S1, S3, S1&S3 for this feature and someone suggested if S1&S3 isn't working try just S1 (S1 and S3 are two sleep states with S3 being deeper).
If it still persists,
If you go to control panel - power options you are supposed to be able to control the period of inactivity required before Sleep and Hibernation occur, by setting your own power specifications in advanced settings. Note, the main purpose is to save battery life on laptops (I suppose a second purpose for desktops etc is to save energy, be more green) so you really do WANT it sleeping but I know it's frustrating if it won't wake up.
Finally, mine does respond to the mouse and keyboard and wakes up my screen but other stuff seems to not wake properly because my very next action causes my system to freeze so I have found if my keyboard action to wake the computer is Ctrl-Alt-Delete then it brings up the windows logoff/switch user menu and from there I can switch user and this cycles all my services and my maching does wake up. This isn't great really because I esentially logged off and back on but I figure it is cleaner than pressing my power button for a hard shut down.