We have a bunch of VB6 applications that access two different database servers (both 32-bit windows 2003, one SQL Server 2000, one SQL Server 2005). About every ten minutes or so, we are getting a few errors:

[Microsoft][ODBC SQL Server Driver]Timeout expired [Microsoft][ODBC SQL Server Driver][DBNETLIB]SQL Server does not exist or access denied. [Microsoft][ODBC SQL Server Driver]ConnectionRead()

This is happening on more than a dozen different computers at random times. We also have IP phones that all run through the same network and those are not having any problems. We can also VNC into a users computer and reproduce the error they were getting, but VNC still continues to work. Email also works. It just seems to be an ODBC connection to SQL Server that causes the issue. The errors happen for both of our SQL Servers. We have scoured google, but haven't been able to come up with a solution. Is there anything we can try to diagnose the problem? Is there any fix out there?