The error was being caused by a limit in the number of connections on the website.

 

To confirm this, we checked the HTTPERR Log files located in:

   %SYSTEMROOT%\System32\LogFiles\HTTPERR\

 

The log showed ConnLimit as the cause of the 503 errors.

 

This can be resolved in the IIS Manager via:

 

   Website -> Properties -> Performance -> Connections Limited to: (We went from 50 to 500 to resolve for us. YMMV)