It seems bizarre a user can connect clear across the country, but I can't sitting in the same room. Here's the steps for me to
recreate the error:

1. Start up Query Analyzer and go to File\Connect
2. Use the ... dialog to find the computer I want.
3. Use sa and its password.

Error: Unable to connect to server LS-IMRIE\SILFW:

Server: Msg 17 Level 16, State 1
[Microsoft].[ODBC SQL Server Driver][DBNETLIB]SQL Server does not exist or access denied.

I also reduplicated the error with Windows Authentication, but I expected
that, too.

This site seems to address the same issue, but I'm not quite understanding it:

I can connect to other machines (servers) in the room, but not this particular one.