I ran into the same issue. It seems like it is solved for me. I went through all my code making sure to invoke webResponse.Close() and/or responseStream.Close() for all my HttpWebResponse objects. The documentation indicates that you can close the stream or the HttpWebResponse object. Calling both is not harmful, so I did. Not closing the responses may cause the application to run out of connections for reuse, and this seems to affect the HttpWebRequest.GetRequestStream as far as I can observe in my code.
Note: or use “using” instead.