I don't know what a "Linksys home box" is - a "router" perhaps?
In general terms, the "Limited or No Connectivity" error means that the network adapter, in this case the wireless adapter in your upstairs computer, has been unable to get its IP address from the DHCP server (the router if that's what you are connecting to) and so Windows has allocated its address instead. When Windows does this, it gets an address in the 169.254.x.x range, which results in Limited or No Connectivity, hence the error.
When first switching on, the "Linksys home box" should be running before the PC is booted, so that (1) the wireless adapter can connect to the network and (2) the wireless adapter can get its IP address.
If you are using Windows XP, what you can do is to open the Network Connections folder, double click the "Wireless Network Connection" and select the Support tab and click the Repair button. This should force the adapter to renew its IP address and, if it can find the DHCP server, it should then get its proper address, probably in the 192.168.x.x range.