.Net C# RESTSharp 10 Minute Timeout
It is possible you are hitting the connection limit for .Net apps, as in MS docs:
"By default, an application using the HttpWebRequest class uses a maximum of two persistent connections to a given server, but you can set the maximum number of connections on a per-application basis."
(https://docs.microsoft.com/en-us/dotnet/framework/network-programming/managing-connections).
Closing the connections should help, or you might be able to increase that limit, that is also in the doc