Problem when connecting to SQL Server
This answer is intentionally Community Wiki to encourage others to edit in other useful links and information for troubleshooting connection problems.

Please try:


Troubleshooting: Timeout Expired
How to Troubleshoot Connecting to the SQL Server Database Engine


Both are Microsoft TechNet guides targeted at your type of problem.


Aaron Bertrand wrote a blog post listing many different error codes for login related connection errors


The Stack Overflow Q &amp; A Help troubleshooting SqlException: Timeout expired on connection, in a non-load situation provides lot of good pointers to check on a box facing this type of issue.