|Both of you....|
Your problem probably has nothing to do with you being on the net at the time this happens, if that applies to both of you.
It sounds like your computer is going into Standby mode after the computer has detected no user inactivity from you for some period of time, but Standby is not working properly, so you can't get out of that no video mode after you have entered it.
E.g. When you are reading something or typing something while on the web, but you aren't changing the web page you're on, that's not perceived as you actively using the computer.
If you have re-loaded your operating system from scratch, you must load the main chipset drivers for the mboard after Setup has finished, otherwise, amongst other things, the operating system may not have the proper info about the ACPI features of the mboard - Standby, Hibernate, Wake on ...., etc., sometimes also Restart, may not work properly.
Similarly, you can have your problem if the main chipset drivers were installed but the ACPI info has become corrupted.
If your mboard has a Via, Intel, or AMD main chipset, it is usually easy to load or re-load your main chipset drivers, and you don't need to look for and un-install possible existing main chipset drivers that have been already installed, before you install them or install them again.
If you mboard has a NVidia main chipset, that's a different matter.
In many cases you can't get a relatively smaller download that is only the main chipset drivers - the main chipset drivers are usually included in the contents of a huge download. If you have already installed such a NVidia download, or installation from a CD, if the ACPI info has become corrupted, you need to un-install the NVidia mboard related software in Control Panel - Add or Remove Programs in XP and below, or in Control Panel - Programs and Features in Vista or Windows 7, then install the huge NVidia software download or huge installation from a CD.
If you install Vista or Windows 7 on an older system that did not originally come with it, not all older mboards will have their ACPI features supported properly by Vista or Windows 7, and in that case Standby can't work properly.
When your problem is caused by the operating system not having the proper ACPI info for you mboard, as a temporary or permanent fix, you can go to Control Panel - Power (or similar) - and set Standby to NEVER come on.
You can set the monitor and/or the hard drive(s) to shut off after xx time of inactivity there - it / they will usually come back on when you move a mouse or press a key, no problem.
Whenever you load Windows from a regular Windows CD (or DVD) from scratch, after Setup is finished you must load the drivers for the mboard, particularly the main chipset drivers, in order for Windows to have the proper drivers for and information about your mboard hardware, including it's AGP or PCI-E, ACPI, USB 2.0 if it has it, and hard drive controller support. If you have a generic system and have the CD that came with the mboard, all the necessary drivers are on it. If you load drivers from the web, brand name system builders and mboard makers often DO NOT have the main chipset drivers listed in the downloads for your model - in that case you must go to the maker of the main chipset's web site, get the drivers, and load them.