If your only concern is the time it takes to receive the response, and not the bandwidth being used, you could get the response asynchronously.
http://msdn.microsoft.com/en-us/library/system.net.httpwebrequest.begingetresponse.aspx
The example given is a bit complicated, but the general idea is that your program will not wait for the response to be downloaded when BeginGetResponse is called, like it would if you just called GetResponse. The first method that you pass to BeginGetResponse is the name of a method (called a "callback") that will get called when the response eventually is fully downloaded. This is where you'd put your code to check the HTTP response code, assuming you cared about that. The 2nd parameter is a "state" object that gets passed to your callback method. We'll use this to make sure everything gets cleaned up properly.
It would look something like this:
private void YourMethod()
{
// Set up your request as usual.
request.BeginGetResponse(DownloadComplete, request);
// Code down here runs immediately, without waiting for the response to download
}
private static void DownloadComplete(IAsyncResult ar)
{
var request = (HttpWebRequest)ar.AsyncState;
var response = request.EndGetResponse(ar);
// You can check your response here to make sure everything worked.
}