Поскольку evt.Error - это исключение WebException (а не исключение ванили), вот что я делаю (прошу прощения за VB.NET):
''' <summary>
''' Extends a WebException to include any body text from the HTTP Response in the .Message
''' </summary>
Friend Function ExtendWebExceptionInfo(ex As Exception) As Exception
Dim wEx As WebException = TryCast(ex, WebException)
If wEx Is Nothing Then Return ex
Dim exMessage As String = Nothing
Using reader As New StreamReader(wEx.Response.GetResponseStream, System.Text.Encoding.UTF8)
exMessage = reader.ReadToEnd
End Using
If Not String.IsNullOrWhiteSpace(exMessage) Then
exMessage = String.Format("{0}{1}{1}The server says:{1}{2}", wEx.Message, vbCrLf, exMessage)
Return New WebException(exMessage, wEx, wEx.Status, wEx.Response)
End If
Return wEx
End Function