Not sure what you're using to download, but have you tried shutting down X (the GUI) and letting the downloads run in text mode?
If it works, it might mean you have a GUI related problem, or the cessation of activity is related to the X server or some module it loads or something it uses to run.
At least, this is what I'd try first in your situation. Stopping the GUI eliminates a lot of memory use and processor time being used to... well, run the GUI
Also, I once had this type of behaviour with a bad DRAM chip - you SURE your hardware is 100% fine??