diff --git a/test/test_download.py b/test/test_download.py
index 787013c34..ee53efa1c 100755
--- a/test/test_download.py
+++ b/test/test_download.py
@@ -105,11 +105,11 @@ def generator(test_case, tname):
             info_dict = tc.get('info_dict', {})
             params = tc.get('params', {})
             if not info_dict.get('id'):
-                raise Exception('Test definition incorrect. \'id\' key is not present')
+                raise Exception(f'Test {tname} definition incorrect - "id" key is not present')
             elif not info_dict.get('ext'):
                 if params.get('skip_download') and params.get('ignore_no_formats_error'):
                     continue
-                raise Exception('Test definition incorrect. The output file cannot be known. \'ext\' key is not present')
+                raise Exception(f'Test {tname} definition incorrect - "ext" key must be present to define the output file')
 
         if 'skip' in test_case:
             print_skipping(test_case['skip'])
@@ -161,7 +161,9 @@ def generator(test_case, tname):
                         force_generic_extractor=params.get('force_generic_extractor', False))
                 except (DownloadError, ExtractorError) as err:
                     # Check if the exception is not a network related one
-                    if not err.exc_info[0] in (urllib.error.URLError, socket.timeout, UnavailableVideoError, http.client.BadStatusLine) or (err.exc_info[0] == urllib.error.HTTPError and err.exc_info[1].code == 503):
+                    if (err.exc_info[0] not in (urllib.error.URLError, socket.timeout, UnavailableVideoError, http.client.BadStatusLine)
+                            or (err.exc_info[0] == urllib.error.HTTPError and err.exc_info[1].code == 503)):
+                        err.msg = f'{getattr(err, "msg", err)} ({tname})'
                         raise
 
                     if try_num == RETRIES: