This is one of those difficult ones to diagnose as it can be hardware or software, primarily drivers. Hardware suspects would be:
1.) Memory, which you have ruled out already
2.) Power supply
3.) CPU temp.

Software suspects would be:
1.) Driver for video card
2.) Driver for NIC
3.) Could be other drivers, but the first two are the most common.

If this were my system, I would start with the driver issue as there is no cost involved, just time. If this didn't resolve the issue, I would open the system, clean (blow) out the power supply and fans, and restart. If the problem goes away, well the answer is obvious, you were dealing with a heat/dust issue. If it returns, but runs much longer, I would either replace the cpu fan or clean and reapply the thermal compound between the CPU and fan assembly. These are all simple and cheap things that can be done to help isolate the problem. A can of compressed air is approx. 5-5 bucks. A new fan would be around 15 bucks, and thermal compound is quite cheap also. Hopefully either the driver issue or one of these hardware tips will restore your system to 100% functionality. Please post back and let us know how you made out. Best of luck.