I tried your code and added some output to it.
        using (var webClient = new System.Net.WebClient())
        {
            Stopwatch timer = Stopwatch.StartNew();
            String url = "http://bg2.cba.pl/realmIP.txt";
            timer.Stop();
            TimeSpan timespan = timer.Elapsed;
            String tex1 = String.Format("{0:00}:{1:00}:{2:00}", timespan.Minutes, timespan.Seconds, timespan.Milliseconds / 10);
            timer = Stopwatch.StartNew();
            String result = webClient.DownloadString(url); // slow as hell
            timespan = timer.Elapsed;
            String tex2 = String.Format("{0:00}:{1:00}:{2:00}", timespan.Minutes, timespan.Seconds, timespan.Milliseconds / 10);
            timer = Stopwatch.StartNew();
            Stream stream = webClient.OpenRead(url);
            timespan = timer.Elapsed;
            String tex3 = String.Format("{0:00}:{1:00}:{2:00}", timespan.Minutes, timespan.Seconds, timespan.Milliseconds / 10);
            timer = Stopwatch.StartNew();
            byte[] result2 = new byte[12];
            int val = webClient.OpenRead(url).Read(result2, 0, 12); // even slower
            timespan = timer.Elapsed;
            String tex4 = String.Format("{0:00}:{1:00}:{2:00}", timespan.Minutes, timespan.Seconds, timespan.Milliseconds / 10);
            textBox1.Text = result;
            t1.Text = tex1;
            t2.Text = tex2;
            t3.Text = tex3;
            t4.Text = tex4;
        }
with the following Result

Your code seems to be okay.
Check your Firewall and all the stuff that is involved