diff --git a/mne/annotations.py b/mne/annotations.py
index 5606ff26fad..845a8b7228f 100644
--- a/mne/annotations.py
+++ b/mne/annotations.py
@@ -278,12 +278,19 @@ class Annotations:
     def __init__(self, onset, duration, description, orig_time=None, ch_names=None):
         self._orig_time = _handle_meas_date(orig_time)
         if isinstance(orig_time, str) and self._orig_time is None:
-            warnings.warn(
-                "The format of the `orig_time` string is not recognised. It must "
-                "conform to the ISO8601 format with at most microsecond precision and "
-                "where the delimiter between date and time is ' '.",
-                RuntimeWarning,
-            )
+            try:  # only warn if `orig_time` is not the default '1970-01-01 00:00:00'
+                if _handle_meas_date(0) == datetime.strptime(
+                    orig_time, "%Y-%m-%d %H:%M:%S"
+                ).replace(tzinfo=timezone.utc):
+                    pass
+            except ValueError:  # error if incorrect datetime format AND not the default
+                warn(
+                    "The format of the `orig_time` string is not recognised. It "
+                    "must conform to the ISO8601 format with at most microsecond "
+                    "precision and where the delimiter between date and time is "
+                    "' '.",
+                    RuntimeWarning,
+                )
         self.onset, self.duration, self.description, self.ch_names = _check_o_d_s_c(
             onset, duration, description, ch_names
         )
diff --git a/mne/tests/test_annotations.py b/mne/tests/test_annotations.py
index c02d37bc92a..55f97360916 100644
--- a/mne/tests/test_annotations.py
+++ b/mne/tests/test_annotations.py
@@ -76,6 +76,9 @@ def windows_like_datetime(monkeypatch):
 
 def test_basics():
     """Test annotation class."""
+    pytest.importorskip("pandas")
+    import pandas as pd
+
     raw = read_raw_fif(fif_fname)
     assert raw.annotations is not None
     assert len(raw.annotations.onset) == 0
@@ -95,6 +98,17 @@ def test_basics():
             assert isinstance(annot.orig_time, datetime)
             assert annot.orig_time.tzinfo is timezone.utc
 
+    # Test bad format `orig_time` str -> `None` raises warning
+    with pytest.warns(
+        RuntimeWarning, match="The format of the `orig_time` string is not recognised."
+    ):
+        bad_orig_time = (
+            pd.Timestamp(_ORIG_TIME)
+            .astimezone(None)
+            .isoformat(sep=" ", timespec="nanoseconds")
+        )
+        Annotations(onset, duration, description, bad_orig_time)
+
     pytest.raises(ValueError, Annotations, onset, duration, description[:9])
     pytest.raises(ValueError, Annotations, [onset, 1], duration, description)
     pytest.raises(ValueError, Annotations, onset, [duration, 1], description)