I agree with Richard. If it's not related to issue outlined in the KB, you might check the URL being used by your connection file and make sure it's not using a public-facing URL. If it is, and you lose connection to the internet, it will fail. In the situation that you are still connected to a network, but that network has no access to the internet, the connection should use internal URLs so that it doesn't attempt to leave your local network to reach the ArcGIS Server. In the event that the machine is completely disconnected from a local network and the internet, you'll probably need to use localhost in the connection URL unless you resolve the machine name to 127.0.0.1 (local loopback) in the hosts file (C:\Windows\System32\drivers\etc). Either way, Fiddler would be helpful in showing you what paths are being used to access your server.