WebException ¿cómo obtener una respuesta completa con un cuerpo?

En WebException no puedo ver el cuerpo de GetResponse. Este es mi código en C #:

try {                
  return GetResponse(url + "." + ext.ToString(), method, headers, bodyParams);
} catch (WebException ex) {
    switch (ex.Status) {
      case WebExceptionStatus.ConnectFailure:
         throw new ConnectionException();                        
     case WebExceptionStatus.Timeout:
         throw new RequestTimeRanOutException();                     
     case WebExceptionStatus.NameResolutionFailure:
         throw new ConnectionException();                        
     case WebExceptionStatus.ProtocolError:
          if (ex.Message == "The remote server returned an error: (401) unauthorized.") {
              throw new CredentialsOrPortalException();
          }
          throw new ProtocolErrorExecption();                    
     default:
          throw;
    }

Veo el encabezado pero no veo el cuerpo. Esta es la salida de Wireshark para la solicitud:

POST /api/1.0/authentication.json HTTP/1.1    
Content-Type: application/x-www-form-urlencoded    
Accept: application/json    
Host: nbm21tm1.teamlab.com    
Content-Length: 49    
Connection: Keep-Alive    

userName=XXX&password=YYYHTTP/1.1 500 Server error    
Cache-Control: private, max-age=0    
Content-Length: 106    
Content-Type: application/json; charset=UTF-8    
Server: Microsoft-IIS/7.5    
X-AspNet-Version: 2.0.50727    
X-Powered-By: ASP.NET    
X-Powered-By: ARR/2.5

Date: Mon, 06 Aug 2012 12:49:41 GMT    
Connection: close    

{"count":0,"startIndex":0,"status":1,"statusCode":500,"error":{"message":"Invalid username or password."}}

¿Es posible de alguna manera ver el texto del mensaje en WebException? Gracias.

var resp = new StreamReader(ex.Response.GetResponseStream()).ReadToEnd();

dynamic obj = JsonConvert.DeserializeObject(resp);
var messageFromServer = obj.error.message;

try {
 WebClient client = new WebClient();
 client.Encoding = Encoding.UTF8;
 string content = client.DownloadString("https://sandiegodata.atlassian.net/wiki/pages/doaddcomment.action?pageId=524365");
 Console.WriteLine(content);
 Console.ReadKey();
} catch (WebException ex) {
 var resp = new StreamReader(ex.Response.GetResponseStream()).ReadToEnd();
 Console.WriteLine(resp);
 Console.ReadKey();
}

Esto solo mejora las respuestas existentes. He escrito un método que se ocupa de los detalles de lanzar / volver a lanzar con un mensaje mejorado, que incluye el cuerpo de la respuesta:

Aquí está mi código (en Client.cs):

/// <summary>
///     Tries to rethrow the WebException with the data from the body included, if possible. 
///     Otherwise just rethrows the original message.
/// </summary>
/// <param name="wex">The web exception.</param>
/// <exception cref="WebException"></exception>
/// <remarks>
///     By default, on protocol errors, the body is not included in web exceptions. 
///     This solutions includes potentially relevant information for resolving the
///     issue.
/// </remarks>
private void ThrowWithBody(WebException wex) {
    if (wex.Status == WebExceptionStatus.ProtocolError) {
        string responseBody;
        try {
            //Get the message body for rethrow with body included
            responseBody = new StreamReader(wex.Response.GetResponseStream()).ReadToEnd();

        } catch (Exception) {
            //In case of failure to get the body just rethrow the original web exception.
            throw wex;
        }

        //include the body in the message
        throw new WebException(wex.Message + $" Response body: '{responseBody}'", wex, wex.Status, wex.Response);
    }

    //In case of non-protocol errors no body is available anyway, so just rethrow the original web exception.
    throw wex;
}

Lo usa en una cláusula de captura como mostró el OP:

//Execute Request, catch the exception to eventually get the body
try {
    //GetResponse....
    }
} catch (WebException wex) {
    if (wex.Status == WebExceptionStatus.ProtocolError) {
        ThrowWithBody(wex);
    }

    //otherwise rethrow anyway
    throw;
}