Also, you could try swapping cables with Dave for a session, see if yours stays up and his drops (that may involve swapping positions with him if his cable is too short).
I have plenty of PCs laying around older than yours... it's worth consideration that the hardware may be just dying of old age. That PC has a built-in 10/100 NIC. Is the Network cable plugged into the port that is part of the motherboard or an add-on PCI card?
If it is the on-board port, you won't be able to do as Zippy suggests.
You could try re-installing the drivers for the on-board network card - just delete it from Device manager and reboot.
It could be worthwhile purchasing a cheap
($10) ethernet NIC to see if that makes a difference.