Fix insert NaT into tzaware datetime index

closes pandas-dev#16357

pandas/core/indexes/datetimes.py

@@ -1776,7 +1776,8 @@ def insert(self, loc, item):
 
         if isinstance(item, (datetime, np.datetime64)):
             self._assert_can_do_op(item)
-            if not self._has_same_tz(item):
+            if not self._has_same_tz(item) and item is not self._na_value:
+                # GH#16537 allow pd.NaT through
                 raise ValueError(
                     'Passed item and index have different timezone')
             # check freq can be preserved on edge cases

pandas/tests/indexes/datetimes/test_indexing.py

@@ -46,6 +46,14 @@ def test_where_tz(self):
         expected = i2
         tm.assert_index_equal(result, expected)
 
+    @pytest.mark.parametrize('tz', [None, 'UTC', 'US/Eastern'])
+    def test_insert_nat(self, tz):
+        # GH#16537
+        idx = pd.DatetimeIndex(['2017-01-01'], tz=tz)
+        res = idx.insert(0, pd.NaT)
+        expected = pd.DatetimeIndex(['NaT', '2017-01-01'], tz=tz)
+        tm.assert_index_equal(res, expected)
+
     def test_insert(self):
         idx = DatetimeIndex(
             ['2000-01-04', '2000-01-01', '2000-01-02'], name='idx')