diff --git a/pandas/core/arrays/datetimes.py b/pandas/core/arrays/datetimes.py
index 2110f782330fb..c11eea603bc64 100644
--- a/pandas/core/arrays/datetimes.py
+++ b/pandas/core/arrays/datetimes.py
@@ -1852,14 +1852,20 @@ def objects_to_datetime64ns(
             yearfirst=yearfirst,
             require_iso8601=require_iso8601,
         )
-    except ValueError as e:
+    except ValueError as err:
         try:
             values, tz_parsed = conversion.datetime_to_datetime64(data)
             # If tzaware, these values represent unix timestamps, so we
             #  return them as i8 to distinguish from wall times
             return values.view("i8"), tz_parsed
         except (ValueError, TypeError):
-            raise e
+            if "Unknown string format" in err.args[0]:
+                raise ValueError(
+                    f"Unexpected value {err.args[1]}.\n"
+                    "You can coerce to NaT by passing `errors='coerce'`"
+                ) from err
+
+            raise err
 
     if tz_parsed is not None:
         # We can take a shortcut since the datetime64 numpy array
diff --git a/pandas/core/tools/timedeltas.py b/pandas/core/tools/timedeltas.py
index d7529ec799022..48be7abc19dee 100644
--- a/pandas/core/tools/timedeltas.py
+++ b/pandas/core/tools/timedeltas.py
@@ -112,9 +112,12 @@ def _coerce_scalar_to_timedelta_type(r, unit="ns", errors="raise"):
     """Convert string 'r' to a timedelta object."""
     try:
         result = Timedelta(r, unit)
-    except ValueError:
+    except ValueError as err:
         if errors == "raise":
-            raise
+            raise ValueError(
+                f"Unexpected value {err.args[0]}.\n"
+                "You can coerce to NaT by passing `errors='coerce'`"
+            ) from err
         elif errors == "ignore":
             return r
 
diff --git a/pandas/tests/tools/test_to_datetime.py b/pandas/tests/tools/test_to_datetime.py
index a91c837c9d9a2..d36e9782624fd 100644
--- a/pandas/tests/tools/test_to_datetime.py
+++ b/pandas/tests/tools/test_to_datetime.py
@@ -1630,7 +1630,10 @@ def test_string_na_nat_conversion(self, cache):
         malformed = np.array(["1/100/2000", np.nan], dtype=object)
 
         # GH 10636, default is now 'raise'
-        msg = r"Unknown string format:|day is out of range for month"
+        msg = (
+            "Unexpected value 1/100/2000.\n"
+            "You can coerce to NaT by passing `errors='coerce'`"
+        )
         with pytest.raises(ValueError, match=msg):
             to_datetime(malformed, errors="raise", cache=cache)
 
diff --git a/pandas/tests/tools/test_to_timedelta.py b/pandas/tests/tools/test_to_timedelta.py
index e3cf3a7f16a82..e8f42c4b9db53 100644
--- a/pandas/tests/tools/test_to_timedelta.py
+++ b/pandas/tests/tools/test_to_timedelta.py
@@ -121,6 +121,13 @@ def test_to_timedelta_invalid(self):
             invalid_data, to_timedelta(invalid_data, errors="ignore")
         )
 
+        msg = (
+            "Unexpected value unit abbreviation w/o a number.\n"
+            "You can coerce to NaT by passing `errors='coerce'`"
+        )
+        with pytest.raises(ValueError, match=msg):
+            pd.to_timedelta("foo", errors="raise")
+
     def test_to_timedelta_via_apply(self):
         # GH 5458
         expected = Series([np.timedelta64(1, "s")])