diff --git a/doc/source/whatsnew/v0.23.1.txt b/doc/source/whatsnew/v0.23.1.txt
index a071d7f3f5534..99e968d908182 100644
--- a/doc/source/whatsnew/v0.23.1.txt
+++ b/doc/source/whatsnew/v0.23.1.txt
@@ -78,6 +78,8 @@ Indexing
 
 - Bug in :meth:`Series.reset_index` where appropriate error was not raised with an invalid level name (:issue:`20925`)
 - Bug in :func:`interval_range` when ``start``/``periods`` or ``end``/``periods`` are specified with float ``start`` or ``end`` (:issue:`21161`)
+- Bug in :meth:`MultiIndex.set_names` where error was raised for a MultiIndex with self.nlevels == 1 (:issue: `21149`)
+- Bug in comparison operations for :class:`MultiIndex` where error was raised on equality / inequality comparison involving a MultiIndex with self.nlevels == 1 (:issue: `21149`)
 -
 
 I/O
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index f79288c167356..2a02e901b1002 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -96,10 +96,13 @@ def cmp_method(self, other):
         if needs_i8_conversion(self) and needs_i8_conversion(other):
             return self._evaluate_compare(other, op)
 
+        from .multi import MultiIndex
         if is_object_dtype(self) and self.nlevels == 1:
-            # don't pass MultiIndex
-            with np.errstate(all='ignore'):
-                result = ops._comp_method_OBJECT_ARRAY(op, self.values, other)
+            if not isinstance(self, MultiIndex):
+                # don't pass MultiIndex
+                with np.errstate(all='ignore'):
+                    result = ops._comp_method_OBJECT_ARRAY(op,
+                                                           self.values, other)
 
         else:
 
@@ -1383,8 +1386,8 @@ def set_names(self, names, level=None, inplace=False):
                    labels=[[0, 0, 1, 1], [0, 1, 0, 1]],
                    names=[u'baz', u'bar'])
         """
-
-        if level is not None and self.nlevels == 1:
+        from .multi import MultiIndex
+        if level is not None and not isinstance(self, MultiIndex):
             raise ValueError('Level must be None for non-MultiIndex')
 
         if level is not None and not is_list_like(level) and is_list_like(
diff --git a/pandas/tests/indexes/test_multi.py b/pandas/tests/indexes/test_multi.py
index c9f6bc9151d00..9c45c8188c3df 100644
--- a/pandas/tests/indexes/test_multi.py
+++ b/pandas/tests/indexes/test_multi.py
@@ -3291,3 +3291,37 @@ def test_duplicate_multiindex_labels(self):
         with pytest.raises(ValueError):
             ind.set_levels([['A', 'B', 'A', 'A', 'B'], [2, 1, 3, -2, 5]],
                            inplace=True)
+
+    testdata = [
+        (pd.MultiIndex.from_product([[0, 1], [1, 0]]), pd.Series(range(4)), 4),
+        (pd.MultiIndex.from_product([[0, 1]]), pd.Series(range(2)), 2),
+    ]
+
+    @pytest.mark.parametrize("midx,idx,count", testdata)
+    def test_multiindex_compare(self, midx, idx, count):
+        # GH 21149 - change in 'def cmp_method()'
+        expected = pd.Series([True]).repeat(count)
+        expected.reset_index(drop=True, inplace=True)
+        result = pd.Series(midx == midx)
+        tm.assert_series_equal(result, expected)
+        result = (idx == idx)
+        tm.assert_series_equal(result, expected)
+
+        expected = pd.Series([False]).repeat(count)
+        expected.reset_index(drop=True, inplace=True)
+        result = pd.Series(midx > midx)
+        tm.assert_series_equal(result, expected)
+        result = pd.Series(midx == idx)
+        tm.assert_series_equal(result, expected)
+
+        with tm.assert_raises_regex(TypeError, 'not supported'):
+            midx > idx
+
+    def test_multiindex_set_names(self):
+        # GH 21149 - change in 'def set_names()'
+        result = pd.MultiIndex.from_product([[0, 1]])
+        result.set_names('first', level=0, inplace=True)
+        expected = pd.MultiIndex(levels=[[0, 1]],
+                                 labels=[[0, 1]],
+                                 names=['first'])
+        tm.assert_index_equal(result, expected)