можно запустить несколько потоков, загрузить файл частями и собрать его.
https://github.com/yingyixu/android-download-manager
Правда автор Китаец, что накладывает некоторые ограничения

... потому как варриант, что -то вроде:
[syntax=java5]
public class uploadFile extends AsyncTask<Void, Void, Boolean> {
private final ProgressDialog dialog = new ProgressDialog(parentActivity);
protected void onPreExecute() {
this.dialog.setMessage("Uploading file");
this.dialog.setCancelable(false);
this.dialog.show();
}
@Override
protected Boolean doInBackground(Void... arg0) {
try {
HttpClient httpClient = new DefaultHttpClient();
HttpPost postRequest = new HttpPost(URLS.PRESCRIPTION_POST_URL);
MultipartEntity reqEntity = new MultipartEntity(
HttpMultipartMode.BROWSER_COMPATIBLE);
reqEntity.addPart("title", new StringBody("This is a title of video file"));
try {
File f = new File(Environment.getExternalStorageDirectory(), "your file name with extension");
FileBody body = new FileBody(f);
reqEntity.addPart("parameter that server will read", body);
} catch (Exception e) {
reqEntity.addPart("parameter that server will read", new StringBody(""));
}
reqEntity.addPart("description", new StringBody("description"));
postRequest.setEntity(reqEntity);
HttpResponse response = httpClient.execute(postRequest);
BufferedReader reader = new BufferedReader(new InputStreamReader(response.getEntity().getContent(), "UTF-8"));
String sResponse; StringBuilder s = new StringBuilder();
while ((sResponse = reader.readLine()) != null) {
s = s.append(sResponse);
}
Log.v("Response for POst", s.toString());
return true;
} catch (Exception e) {
Log.e("MyPharmacyOptions", "Error :: " + e);
}
return false;
}
@Override
protected void onPostExecute(Boolean result) {
if (this.dialog.isShowing()) {
this.dialog.dismiss();
}
if (result) {
Toast.makeText(parentActivity,
"File uploaded successfully", Toast.LENGTH_LONG)
.show();
} else {
Toast.makeText(parentActivity, "Your Request not complete",
Toast.LENGTH_LONG).show();
}
}
}
[/syntax]