Hi Peter,
Thanks for reposting this; this error is basically what you can expect from a "network stack" being overloaded; each request to a ProGet feed can potentially open another requests to each connector configured (maybe NuGet.org, maybe others), and if you are doing a lot of requests at once, you'll get a lot of network activity queuing up. SQL Server also running on the network, so those just get added to the queue, and eventually you run out of connection.
Do you have a lot of containers running on the same computer that ProGet is running on? If so, that coudl also contribute to the network stack being overloaded.
The best way to solve is with more hardware, or by removing connectors to nuget.org, etc.
Also note that users can overload a server as well, and this is why load balancing will be very important if you want to keep reliability; See How to Prevent Server Overload in ProGet to learn more.