diff --git a/pandas/tests/frame/test_operators.py b/pandas/tests/frame/test_operators.py
index 19d91241d6a6b..39f738657ddd5 100644
--- a/pandas/tests/frame/test_operators.py
+++ b/pandas/tests/frame/test_operators.py
@@ -218,6 +218,42 @@ def test_logical_with_nas(self):
         expected = Series([True, True])
         tm.assert_series_equal(result, expected)
 
+    @pytest.mark.parametrize(
+        "left, right, op, expected",
+        [
+            (
+                [True, False, np.nan],
+                [True, False, True],
+                operator.and_,
+                [True, False, False],
+            ),
+            (
+                [True, False, True],
+                [True, False, np.nan],
+                operator.and_,
+                [True, False, False],
+            ),
+            (
+                [True, False, np.nan],
+                [True, False, True],
+                operator.or_,
+                [True, False, False],
+            ),
+            (
+                [True, False, True],
+                [True, False, np.nan],
+                operator.or_,
+                [True, False, True],
+            ),
+        ],
+    )
+    def test_logical_operators_nans(self, left, right, op, expected):
+        # GH 13896
+        result = op(DataFrame(left), DataFrame(right))
+        expected = DataFrame(expected)
+
+        tm.assert_frame_equal(result, expected)
+
 
 class TestDataFrameOperators:
     @pytest.mark.parametrize(
diff --git a/pandas/tests/series/test_operators.py b/pandas/tests/series/test_operators.py
index 7d212ee7cd667..983560d68c28c 100644
--- a/pandas/tests/series/test_operators.py
+++ b/pandas/tests/series/test_operators.py
@@ -43,6 +43,42 @@ def test_logical_operators_bool_dtype_with_empty(self):
         expected = s_tft
         tm.assert_series_equal(res, expected)
 
+    @pytest.mark.parametrize(
+        "left, right, op, expected",
+        [
+            (
+                [True, False, np.nan],
+                [True, False, True],
+                operator.and_,
+                [True, False, False],
+            ),
+            (
+                [True, False, True],
+                [True, False, np.nan],
+                operator.and_,
+                [True, False, False],
+            ),
+            (
+                [True, False, np.nan],
+                [True, False, True],
+                operator.or_,
+                [True, False, False],
+            ),
+            (
+                [True, False, True],
+                [True, False, np.nan],
+                operator.or_,
+                [True, False, True],
+            ),
+        ],
+    )
+    def test_logical_operators_nans(self, left, right, op, expected):
+        # GH 13896
+        result = op(Series(left), Series(right))
+        expected = Series(expected)
+
+        tm.assert_series_equal(result, expected)
+
     def test_logical_operators_int_dtype_with_int_dtype(self):
         # GH#9016: support bitwise op for integer types