Ok, watch your voltages while you run edonkey and see if they fluctuate.
But it's interesting that the reboots only occur while your doing online stuff. Do you do any other heavy duty online stuff? Try playing some other online games and seeing if you get the reboots. Or do something else that uses a lot of bandwidth. Also, try doing something that doesn't use the net but that pegs your cpu at full load. What happens then?
Given your description I'm inclined to think that it's not a power problem. That's just a gut reaction, but you should be testing your system by using it in different ways and seeing if you can isolate when the reboots occur.