In addition to the excellent advice by Macrovision (and this is really just an adjunct to what he said), you should take a can of compressed air and clean out all the dust...particularly around the cpu heat sink and all the fans...cpu, psu (power supply unit), video card and case.
A cheap (as in free), effective and easy method to test your RAM is to use memtest86.
Here is the download site for the freeware version. You can also purchase a disc, but it's easy to make your own. You can also run it from a floppy.
I'd recommend that you take a look at the Technical Info on the homepage,
Memtest86 will work with any OS and actually will work without an OS at all.
You can also just pull out a stick of RAM, one at a time and see how your rig runs, but this is hardly scientific...still, it may give you your answer.
It's also a good idea to pull out the RAM, clean-out the sockets with compressed air and re-seat the RAM.
I think he meant for you to do this as well (implied)...since it's part of troubleshooting the RAM