If you create a database connection to an external server, and test it, obviously you know that the server you are testing is the server that is down.
Unfortunately, if that server goes down after everything is up and working, and you attempt to publish, you will get an extremely cryptic error message that forces you to assume there is a problem with your local server.
 Internal Error: Invalid compiler output.
A network-related or instance-specific error occurred while establishing a connection to SQL Server. The server was not found or was not accessible. Verify that the instance name is correct and that SQL Server is configured to allow remote connections. (provider: TCP Provider, error: 0 - A connection attempt failed because the connected party did not properly respond after a period of time, or established connection failed because connected host has failed to respond.)
No where in that error message does Service Studio (or even Service Center if compiling from there) nor even in Lifetime does that error message tell you that the SQL problem is not with your local machine, but with an external SQL server.
Please include hostnames of the SQL server in question with the error messages, so that we know WHAT connection you are referring to. It will save people HOURS of work.