We have 4 API servers and 2 web servers that connect to a SQL 2008 R2 server. Recently we've noticed that the web servers sometimes can't find the SQL server. This doesn't happen for every connection, but it does happen several times per minute. All the servers are on Windows 2008 R2, and connect to the DB via TCP. We've verified that the DB is reachable, and in fact, the API servers don't have any problems connecting to the DB.
Because the connection problems are intermittent it is proving difficult to troubleshoot. We've verified that the firewall is not blocking the connection, and that the maximum number of connections is not being reached (max is the default of 32767, and our average connection count at any given time is around 1000).
What might be causing these irregular connection problems?