These errors suggest that the appserver is hitting limitations when trying to establish new TCP connections
"If you run netstat -n and you see that close to 4000 connections to the IP address of the target computer that is running SQL Server are in a TIME_WAIT state, you can both increase the default MaxUserPort setting and reduce the TcpTimedWaitDelay setting so that you do not run out of client anonymous ports. For example, you can set the MaxUserPort setting to 20000 and set the TcpTimedWaitDelay setting to 30. A lower TcpTimedWaitDelay setting means that the sockets wait in the TIME_WAIT state for less time. A higher MaxUserPort setting means that you can have more sockets in the TIME_WAIT state.
Note that if you adjust the MaxUserPort or TcpTimedWaitDelay setting, you must restart Microsoft Windows for the new setting to take effect. The MaxUserPort and TcpTimedWaitDelay settings are for any client computer that is talking to a computer that is running SQL Server over TCP/IP sockets. These settings do not have any effect if they are set on the computer that is running SQL Server unless you are making local TCP/IP socket connections to the local computer that is running SQL Server"
The default for MaxUserPort is only 5000. Increase that to 10000 or 15000.
The default for TCPTimedWaitDelay is 240 seconds. Teduce that to 30.
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You could just try to restart the appserver and free some ports. And try again.