From the description, it sounds like the computer is having an issue with power managment or, believe it or not, a screen saver.
I would check the settings for your screen saver (if you are using one) and the settings for your power and hibernation options. Since your issue only happens when your computer is sitting idle for a period of time, it seems like those are the likely culprits.
It seems as if your computer is set in the power options to go into sleep mode or hibernation mode after a certain period of inactivity... say the 10 or 20 minute timeframe you are experiencing. Some glitch in the process of entering the power saving mode is causing the computer to crash. I know you say you disabled both... but I would check under the advanced settings for your power saver options... many times, there is an advanced option for the hibernation mode that is unaffected by the general settings. Make sure you turn OFF the hibernation mode entirely in the advanced settings.
Also, you don't state what brand of computer this is... check to make sure they do not have their own, custom power control options that might be overriding the general Windows settings.
Other things to look at that could be causing the issue:
Windows Indexing. Windows 7 computers have drive file indexing turned on by default. This process usually lies dormant on your computer while in use... but during "down times" when the computer sense it is being lightly used, the file indexing will fire up and attempt to index the drives. If the process is corrupt, it could be causing your computer to crash.
A third-party scheduled application. Do you have an antivirus application on your computer? There are many third-party programs that also attempt to use a computer's "down time" to run some background process. Perhaps one or more of these applications are causing the computer to crash.
Virus or malware infection. Just like legitimate third-party apps, there are a ton of malware infections that will infect a computer and lie dormant in the background until the computer becomes idle. These applications may then start to run some process while the computer is lightly used (usually to start e-mail spamming or downloading other spyware) and these applications can cause problems like you are seeing.