Nice troublshooting job.. Looks like you have narrowed it down to the computer having the problem.. I would make sure that you have call waiting disabled in your connection/modem settings.
Also lower the data transfer speed and make sure you don't have the modem set to disconnect after a certain period of in-activity.
I used to know a few strings to enter but I haven't used a modem in 2 years. You know the saying "If you don't use it you lose it". Well, I lost it. When all else fails contact your ISP..
Cheers
Digital_DNA