Unless it's a memory problem or something (in other words, unless other programs do it too), then you can be pretty sure that IE is causing something to mess up badly - remember, IE is linked to the windows main kernel - if it goes down, it usually takes windows with it. Are any of your other programs crashing (but not resetting the computer)?
If everything else seems okay, then try switching IE for an alternative browser, such as Netscape, Mozilla, Opera, etc.
If that doesn't help, it might be a problem with the network connections of your computer. Have you got any other programs that regularly connect to the internet, and do they have problems?
If everything leads to a dead end, it could be a motherboard problem. On an old motherboard (years old) I had, whenever too much tried to happen at once, whenever the graphics card was used to run video, and after a little while anyway, it just shut off. No cure, except a new motherboard. It exhibited your computer's symptoms as well - not wanting to get back up after it went to bed.