diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index 40d3823c9700b..c2edfd53e1207 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -11,7 +11,7 @@
 
 from pandas.core.dtypes.common import _NS_DTYPE, is_float, is_integer, is_scalar
 from pandas.core.dtypes.dtypes import DatetimeTZDtype
-from pandas.core.dtypes.missing import isna
+from pandas.core.dtypes.missing import is_valid_nat_for_dtype, isna
 
 from pandas.core.accessor import delegate_names
 from pandas.core.arrays.datetimes import (
@@ -922,9 +922,14 @@ def insert(self, loc, item):
         -------
         new_index : Index
         """
-        if is_scalar(item) and isna(item):
+        if is_valid_nat_for_dtype(item, self.dtype):
             # GH 18295
             item = self._na_value
+        elif is_scalar(item) and isna(item):
+            # i.e. timedeltat64("NaT")
+            raise TypeError(
+                f"cannot insert {type(self).__name__} with incompatible label"
+            )
 
         freq = None
 
diff --git a/pandas/core/indexes/timedeltas.py b/pandas/core/indexes/timedeltas.py
index ee6e5b984ae7b..59fc53a17590c 100644
--- a/pandas/core/indexes/timedeltas.py
+++ b/pandas/core/indexes/timedeltas.py
@@ -16,7 +16,7 @@
     is_timedelta64_ns_dtype,
     pandas_dtype,
 )
-from pandas.core.dtypes.missing import isna
+from pandas.core.dtypes.missing import is_valid_nat_for_dtype, isna
 
 from pandas.core.accessor import delegate_names
 from pandas.core.arrays import datetimelike as dtl
@@ -397,15 +397,16 @@ def insert(self, loc, item):
         new_index : Index
         """
         # try to convert if possible
-        if _is_convertible_to_td(item):
-            try:
-                item = Timedelta(item)
-            except ValueError:
-                # e.g. str that can't be parsed to timedelta
-                pass
-        elif is_scalar(item) and isna(item):
+        if isinstance(item, self._data._recognized_scalars):
+            item = Timedelta(item)
+        elif is_valid_nat_for_dtype(item, self.dtype):
             # GH 18295
             item = self._na_value
+        elif is_scalar(item) and isna(item):
+            # i.e. datetime64("NaT")
+            raise TypeError(
+                f"cannot insert {type(self).__name__} with incompatible label"
+            )
 
         freq = None
         if isinstance(item, Timedelta) or (is_scalar(item) and isna(item)):
diff --git a/pandas/tests/indexes/datetimes/test_indexing.py b/pandas/tests/indexes/datetimes/test_indexing.py
index ef0d2cd2e48cc..210b28aa0c393 100644
--- a/pandas/tests/indexes/datetimes/test_indexing.py
+++ b/pandas/tests/indexes/datetimes/test_indexing.py
@@ -317,7 +317,9 @@ def test_take_fill_value_with_timezone(self):
 
 
 class TestDatetimeIndex:
-    @pytest.mark.parametrize("null", [None, np.nan, pd.NaT])
+    @pytest.mark.parametrize(
+        "null", [None, np.nan, np.datetime64("NaT"), pd.NaT, pd.NA]
+    )
     @pytest.mark.parametrize("tz", [None, "UTC", "US/Eastern"])
     def test_insert_nat(self, tz, null):
         # GH#16537, GH#18295 (test missing)
@@ -326,6 +328,12 @@ def test_insert_nat(self, tz, null):
         res = idx.insert(0, null)
         tm.assert_index_equal(res, expected)
 
+    @pytest.mark.parametrize("tz", [None, "UTC", "US/Eastern"])
+    def test_insert_invalid_na(self, tz):
+        idx = pd.DatetimeIndex(["2017-01-01"], tz=tz)
+        with pytest.raises(TypeError, match="incompatible label"):
+            idx.insert(0, np.timedelta64("NaT"))
+
     def test_insert(self):
         idx = DatetimeIndex(["2000-01-04", "2000-01-01", "2000-01-02"], name="idx")
 
diff --git a/pandas/tests/indexes/timedeltas/test_indexing.py b/pandas/tests/indexes/timedeltas/test_indexing.py
index 0114dfef548de..b70a3d17a10ab 100644
--- a/pandas/tests/indexes/timedeltas/test_indexing.py
+++ b/pandas/tests/indexes/timedeltas/test_indexing.py
@@ -219,11 +219,29 @@ def test_insert(self):
             assert result.name == expected.name
             assert result.freq == expected.freq
 
+    @pytest.mark.parametrize(
+        "null", [None, np.nan, np.timedelta64("NaT"), pd.NaT, pd.NA]
+    )
+    def test_insert_nat(self, null):
         # GH 18295 (test missing)
+        idx = timedelta_range("1day", "3day")
+        result = idx.insert(1, null)
         expected = TimedeltaIndex(["1day", pd.NaT, "2day", "3day"])
-        for na in (np.nan, pd.NaT, None):
-            result = timedelta_range("1day", "3day").insert(1, na)
-            tm.assert_index_equal(result, expected)
+        tm.assert_index_equal(result, expected)
+
+    def test_insert_invalid_na(self):
+        idx = TimedeltaIndex(["4day", "1day", "2day"], name="idx")
+        with pytest.raises(TypeError, match="incompatible label"):
+            idx.insert(0, np.datetime64("NaT"))
+
+    def test_insert_dont_cast_strings(self):
+        # To match DatetimeIndex and PeriodIndex behavior, dont try to
+        #  parse strings to Timedelta
+        idx = timedelta_range("1day", "3day")
+
+        result = idx.insert(0, "1 Day")
+        assert result.dtype == object
+        assert result[0] == "1 Day"
 
     def test_delete(self):
         idx = timedelta_range(start="1 Days", periods=5, freq="D", name="idx")