diff --git a/zerver/tests/test_queue_worker.py b/zerver/tests/test_queue_worker.py
index 072d66aa8a..9e6e1b8697 100644
--- a/zerver/tests/test_queue_worker.py
+++ b/zerver/tests/test_queue_worker.py
@@ -26,6 +26,7 @@ from zerver.models import (
     PreregistrationUser,
     ScheduledMessageNotificationEmail,
     UserActivity,
+    UserProfile,
     get_client,
     get_realm,
     get_stream,
@@ -345,6 +346,33 @@ class WorkerTest(ZulipTestCase):
             {"where art thou, othello?"},
         )
 
+        with send_mock as sm, timer_mock as tm:
+            with simulated_queue_client(lambda: fake_client):
+                time_zero = datetime.datetime(2021, 1, 1, tzinfo=datetime.timezone.utc)
+                # Verify that we make forward progress if one of the messages throws an exception
+                fake_client.enqueue("missedmessage_emails", hamlet_event1)
+                fake_client.enqueue("missedmessage_emails", hamlet_event2)
+                fake_client.enqueue("missedmessage_emails", othello_event)
+                with patch("zerver.worker.queue_processors.timezone_now", return_value=time_zero):
+                    mmw.setup()
+                    mmw.start()
+
+                def fail_some(user: UserProfile, *args: Any) -> None:
+                    if user.id == hamlet.id:
+                        raise RuntimeError
+
+                sm.side_effect = fail_some
+                one_minute_overdue = expected_scheduled_timestamp + datetime.timedelta(seconds=60)
+                with patch(
+                    "zerver.worker.queue_processors.timezone_now", return_value=one_minute_overdue
+                ), self.assertLogs(level="ERROR") as error_logs:
+                    mmw.maybe_send_batched_emails()
+                    self.assertIn(
+                        "ERROR:root:Failed to process 2 missedmessage_emails for user 10",
+                        error_logs.output[0],
+                    )
+                    self.assertEqual(ScheduledMessageNotificationEmail.objects.count(), 0)
+
     def test_push_notifications_worker(self) -> None:
         """
         The push notifications system has its own comprehensive test suite,
diff --git a/zerver/worker/queue_processors.py b/zerver/worker/queue_processors.py
index a9f3a3fdae..5e9de28bf0 100644
--- a/zerver/worker/queue_processors.py
+++ b/zerver/worker/queue_processors.py
@@ -659,7 +659,20 @@ class MissedMessageWorker(QueueProcessingWorker):
                         len(events),
                         user_profile_id,
                     )
-                    handle_missedmessage_emails(user_profile_id, events)
+                    try:
+                        # Because we process events in batches, an
+                        # escaped exception here would lead to
+                        # duplicate messages being sent for other
+                        # users in the same events_to_process batch,
+                        # and no guarantee of forward progress.
+                        handle_missedmessage_emails(user_profile_id, events)
+                    except Exception:
+                        logging.exception(
+                            "Failed to process %d missedmessage_emails for user %s",
+                            len(events),
+                            user_profile_id,
+                            stack_info=True,
+                        )
 
                 events_to_process.delete()