Tak chyba byla jinde.
A to v tom, jak jsem postupně otevíral spojení.

Kdyby to někomu pomohlo, funkce odzkoušená vypadá takto:

private ArrayList<String> Update(final String urlString, String login, String password)
        {
            String response2 = "";
            OutputStream os = null;
            InputStream is = null;
            HttpURLConnection conn = null;
            mujString = new ArrayList<>();
            int resCode = 0;
            try {
                URL url = new URL(urlString);
                conn = (HttpURLConnection) url.openConnection();

                conn.setReadTimeout(10000);
                conn.setConnectTimeout(15000);
                conn.setRequestMethod("POST");
                conn.setDoInput(true);
                conn.setDoOutput(true);

                List<NameValuePair> params = new ArrayList<NameValuePair>();
                params.add(new BasicNameValuePair("login", login));
                params.add(new BasicNameValuePair("password", password));
                //params.add(new BasicNameValuePair("thirdParam", paramValue3));

                os = conn.getOutputStream();

                BufferedWriter writer = new BufferedWriter(new OutputStreamWriter(os, "UTF-8"));

                writer.write(getQuery(params));

                writer.flush();
                writer.close();
                os.close();
                resCode = conn.getResponseCode();

                String line;
                BufferedReader br=new BufferedReader(new InputStreamReader(conn.getInputStream()));
                while ((line=br.readLine()) != null) {
                    response2 += line;
                }

                //mujString.add(String.valueOf(resCode));
                //mujString.add(response2);
            }
            catch (Exception ex) {

            }
            finally {
                conn.disconnect();

            }

            mujString.add(String.valueOf(resCode));
            mujString.add(response2);

            return  mujString;
        }