If the original ram was going bad, adding more won't help. Try running Memtest 86+ for a few hours. It is a program that runs outside of Windows...you boot into the disk or the usb drive. You can find it here: http://www.memtest.org/#downiso Download the iso in zip form, unzip the ISO and then burn it to a blank cd with your normal burning program, or use ImgBurn, which is a free burning program.
Put it in the drive, reboot the machine and let Memtest run for a few hours at least. Overnight is better. You should see NO errors while running Memtest.
If the computer is not set up to boot from a cd or dvd, you'll have to change the boot order in the bios of the computer. Look at your computer manual for instructions on getting in the bios.
The next thing to look at is your temperatures while running the computer. Examine the heatsink on the cpu, and see if it is clogged with dirt. Get a can of compressed air and blow out any dust you can find. Don't let the fans freewheel when you blow air...it can damage the bearings, so turn off the computer and hold the fan blades in place with a finger when blowing out dust.
If you have an Intel processor, you can use Real Temp to monitor temperatures. http://www.techpowerup.com/realtemp/
Do you have any crash reports? Look in Control Panel, Administrative Tools, Event Viewer.