I ran into this many moons ago. Bottom line is you are running out of available ports.
First make sure your calling application has collection pooling on.
If that does then check the number of available ports for the SQL Server.
What is happening is that if pooling is off then every call takes a port and it takes by default 4 minutes to have the port expire, and you are running out of ports.
If pooling is on then you need to profile all the ports of SQL Server and make sure you have enough and expand them if necessary.
When I came across this error, connection pooling was off and it caused this issue whenever a decent load was put on the website. We did not see it in development because the load was 2 or 3 people at max, but once the number grew over 10 we kept seeing this error. We turned pooling on, and it fixed it.