diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 5fe7d12188860..2f4e961ff433f 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -397,6 +397,7 @@ Categorical
 - Bug where :class:`Categorical` comparison operator ``__ne__`` would incorrectly evaluate to ``False`` when either element was missing (:issue:`32276`)
 - :meth:`Categorical.fillna` now accepts :class:`Categorical` ``other`` argument (:issue:`32420`)
 - Bug where :meth:`Categorical.replace` would replace with ``NaN`` whenever the new value and replacement value were equal (:issue:`33288`)
+- Bug where an ordered :class:`Categorical` containing only ``NaN`` values would raise rather than returning ``NaN`` when taking the minimum or maximum  (:issue:`33450`)
 
 Datetimelike
 ^^^^^^^^^^^^
diff --git a/pandas/core/arrays/categorical.py b/pandas/core/arrays/categorical.py
index c9b8db28e0cf6..b3fb3459891e0 100644
--- a/pandas/core/arrays/categorical.py
+++ b/pandas/core/arrays/categorical.py
@@ -2143,7 +2143,7 @@ def min(self, skipna=True):
 
         good = self._codes != -1
         if not good.all():
-            if skipna:
+            if skipna and good.any():
                 pointer = self._codes[good].min()
             else:
                 return np.nan
@@ -2178,7 +2178,7 @@ def max(self, skipna=True):
 
         good = self._codes != -1
         if not good.all():
-            if skipna:
+            if skipna and good.any():
                 pointer = self._codes[good].max()
             else:
                 return np.nan
diff --git a/pandas/tests/arrays/categorical/test_analytics.py b/pandas/tests/arrays/categorical/test_analytics.py
index c470f677b5386..67b491165b8cc 100644
--- a/pandas/tests/arrays/categorical/test_analytics.py
+++ b/pandas/tests/arrays/categorical/test_analytics.py
@@ -88,6 +88,14 @@ def test_min_max_with_nan(self, skipna):
             assert _min == 2
             assert _max == 1
 
+    @pytest.mark.parametrize("function", ["min", "max"])
+    @pytest.mark.parametrize("skipna", [True, False])
+    def test_min_max_only_nan(self, function, skipna):
+        # https://github.com/pandas-dev/pandas/issues/33450
+        cat = Categorical([np.nan], categories=[1, 2], ordered=True)
+        result = getattr(cat, function)(skipna=skipna)
+        assert result is np.nan
+
     @pytest.mark.parametrize("method", ["min", "max"])
     def test_deprecate_numeric_only_min_max(self, method):
         # GH 25303