Check your event viewer in Administrative tools for any clues to the error. Maybe you have a corrupt nic driver or something. Have you tried power cycling the router?

Unplug the router and modem or the combo unit. Shutdown the PC for a minute and restart, wait until everything has loaded and the hard drive goes quiet. Then plug your router and modem back in and let them initialize. Restart the PC and see what happens.

Check your router (look on the bottom for login details) and make sure that both machines aren't being assigned the same IP. If DHCP is off, turn it on, or assign static addresses to each machine. The two machines may have the same preferred IP and conflicts taking place, which explains why the release/renew fix may temporarily work. In theory it shouldn't happen, but anything is possible with computers. If it's not been an issue in the past, perhaps hold down the small reset button on your router until it powercycles and clears all settings.