Is the Linksys a Broadband Router, or truely a switch? Are the IP addresses public (coming from your ISP) or private with your own internal DHCP server?
Also, check the Lease Time when you run WINIPCFG /ALL. Check to see that the time is not hours, but days. If the Lease Time is set to hours, then you have a problem. Well, a problem if your DHCP server is your own. If not, then your ISP has set the Lease Time too low. I would call your ISP and find out.