diff --git a/yt_dlp/YoutubeDL.py b/yt_dlp/YoutubeDL.py
index 9672d0cd3..23e42f740 100644
--- a/yt_dlp/YoutubeDL.py
+++ b/yt_dlp/YoutubeDL.py
@@ -954,13 +954,13 @@ def report_file_delete(self, file_name):
         except UnicodeEncodeError:
             self.to_screen('Deleting existing file')
 
-    def raise_no_formats(self, info, forced=False):
+    def raise_no_formats(self, info, forced=False, *, msg=None):
         has_drm = info.get('__has_drm')
-        msg = 'This video is DRM protected' if has_drm else 'No video formats found!'
-        expected = self.params.get('ignore_no_formats_error')
-        if forced or not expected:
+        ignored, expected = self.params.get('ignore_no_formats_error'), bool(msg)
+        msg = msg or has_drm and 'This video is DRM protected' or 'No video formats found!'
+        if forced or not ignored:
             raise ExtractorError(msg, video_id=info['id'], ie=info['extractor'],
-                                 expected=has_drm or expected)
+                                 expected=has_drm or ignored or expected)
         else:
             self.report_warning(msg)
 
@@ -2440,11 +2440,14 @@ def sanitize_numeric_fields(info):
         if not self.params.get('allow_unplayable_formats'):
             formats = [f for f in formats if not f.get('has_drm')]
 
-        if info_dict.get('is_live'):
-            get_from_start = bool(self.params.get('live_from_start'))
+        get_from_start = not info_dict.get('is_live') or bool(self.params.get('live_from_start'))
+        if not get_from_start:
+            info_dict['title'] += ' ' + datetime.datetime.now().strftime('%Y-%m-%d %H:%M')
+        if info_dict.get('is_live') and formats:
             formats = [f for f in formats if bool(f.get('is_from_start')) == get_from_start]
-            if not get_from_start:
-                info_dict['title'] += ' ' + datetime.datetime.now().strftime('%Y-%m-%d %H:%M')
+            if get_from_start and not formats:
+                self.raise_no_formats(info_dict, msg='--live-from-start is passed, but there are no formats that can be downloaded from the start. '
+                                                     'If you want to download from the current time, pass --no-live-from-start')
 
         if not formats:
             self.raise_no_formats(info_dict)