Could you be having a firewall issue or connection string. You can try disabling the firewall to see if it is causing the problem.
The connection to the database is basically just tcp/ip traffic so if both of your server and client are in the local network, internet doesn't have anything to do with this. Also note that you may have firewall on both sides, at the server and at the client, depending on the OS versions.
Check the connection string that it's properly pointing to the correct server, correct instance name (if any) and with corerct authentication. You can check the connection string using
http://www.connectionstrings.com/[
^]