We even tried to restart the instance into single user mode, however, every time that happened something else would take the connection before we could get into the instance. We eventually restarted the SQL Server instance to normal operation so that we could investigate why we could not get a connection when in single user mode.
Turns out that with the production nature of the instance, the clients large farm of application servers was connecting to it faster than we could. This was discovered by using sp_who2, however, you could use the DMV sys.dm_exec_connections to see what is connecting to the instance if you desired. So, we needed a way to block incoming connections while not being evasive like shutting down the application servers or a large network change.
This is where the brilliance comes in.
Click through for the idea. This is the type of thing you keep in your back pocket in a real pinch, but hope never to need to use.