diff --git a/seaborn/_core/data.py b/seaborn/_core/data.py
index c17bfe95c5..b17d4fee2e 100644
--- a/seaborn/_core/data.py
+++ b/seaborn/_core/data.py
@@ -312,6 +312,12 @@ def convert_dataframe_to_pandas(data: object) -> pd.DataFrame:
         # in Plot.add(). But noting here in case this seems to be a bottleneck.
         return pd.api.interchange.from_dataframe(data)
     except Exception as err:
+        try:
+            # An object's own to_pandas method may be more flexible than the interchange
+            # See https://github.com/mwaskom/seaborn/pull/3534 for more context
+            return data.to_pandas()  # type: ignore[attr-defined]
+        except Exception:
+            pass
         msg = (
             "Encountered an exception when converting data source "
             "to a pandas DataFrame. See traceback above for details."
diff --git a/tests/_core/test_data.py b/tests/_core/test_data.py
index 0e67ed37b4..cf29f98443 100644
--- a/tests/_core/test_data.py
+++ b/tests/_core/test_data.py
@@ -429,6 +429,19 @@ def test_data_interchange_failure(self, mock_long_df):
         with pytest.raises(RuntimeError, match="Encountered an exception"):
             PlotData(mock_long_df, {"x": "x"})
 
+    @pytest.mark.skipif(
+        condition=not hasattr(pd.api, "interchange"),
+        reason="Tests behavior assuming support for dataframe interchange"
+    )
+    def test_data_interchange_failure_with_fallback(self, mock_long_df, long_df):
+
+        mock_long_df._data = None  # Break __dataframe__()
+        mock_long_df.to_pandas = lambda: long_df
+        variables = {"x": "x", "y": "z", "color": "a"}
+        p = PlotData(mock_long_df, variables)
+        for var, col in variables.items():
+            assert_vector_equal(p.frame[var], long_df[col])
+
     @pytest.mark.skipif(
         condition=hasattr(pd.api, "interchange"),
         reason="Tests graceful failure without support for dataframe interchange"