diff --git a/pandas/core/arrays/datetimes.py b/pandas/core/arrays/datetimes.py
index 267780219c74e..d7feb194af11e 100644
--- a/pandas/core/arrays/datetimes.py
+++ b/pandas/core/arrays/datetimes.py
@@ -1952,14 +1952,21 @@ 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
+            # GH#10720. If we failed to parse datetime then notify
+            # that flag errors='coerce' could be used to NaT.
+            # Trying to distinguish exception based on message.
+            if "Unknown string format" in err.args[0]:
+                msg = f"Unknown string format: {err.args[1]}"
+                msg = f"{msg}. You can coerce to NaT by passing errors='coerce'"
+                raise ValueError(msg).with_traceback(err.__traceback__)
+            raise
 
     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 3e185feaea38e..1f205aff6db0a 100644
--- a/pandas/core/tools/timedeltas.py
+++ b/pandas/core/tools/timedeltas.py
@@ -113,9 +113,11 @@ def _coerce_scalar_to_timedelta_type(r, unit="ns", errors="raise"):
 
     try:
         result = Timedelta(r, unit)
-    except ValueError:
+    except ValueError as err:
         if errors == "raise":
-            raise
+            # GH#10720
+            msg = f"{err.args[0]}. You can coerce to NaT by passing errors='coerce'"
+            raise ValueError(msg).with_traceback(err.__traceback__)
         elif errors == "ignore":
             return r
 
diff --git a/pandas/tests/indexes/datetimes/test_tools.py b/pandas/tests/indexes/datetimes/test_tools.py
index a6a43697c36dc..6555d4a049a93 100644
--- a/pandas/tests/indexes/datetimes/test_tools.py
+++ b/pandas/tests/indexes/datetimes/test_tools.py
@@ -480,6 +480,17 @@ def test_to_datetime_unparseable_ignore(self):
         s = "Month 1, 1999"
         assert pd.to_datetime(s, errors="ignore") == s
 
+    def test_to_datetime_unparseable_raise(self):
+        # GH#10720
+        invalid_data = "Month 1, 1999"
+        expected_args = (
+            f"Unknown string format: {invalid_data}. "
+            "You can coerce to NaT by passing errors='coerce'"
+        )
+
+        with pytest.raises(ValueError, match=expected_args):
+            pd.to_datetime(invalid_data, errors="raise")
+
     @td.skip_if_windows  # `tm.set_timezone` does not work in windows
     def test_to_datetime_now(self):
         # See GH#18666
diff --git a/pandas/tests/indexes/timedeltas/test_tools.py b/pandas/tests/indexes/timedeltas/test_tools.py
index 223bde8b0e2c2..75534c752f44a 100644
--- a/pandas/tests/indexes/timedeltas/test_tools.py
+++ b/pandas/tests/indexes/timedeltas/test_tools.py
@@ -111,6 +111,15 @@ def test_to_timedelta_invalid(self):
             invalid_data, to_timedelta(invalid_data, errors="ignore")
         )
 
+        # GH#10720
+        invalid_data = "some_nonesense"
+        expected_msg = (
+            "unit abbreviation w/o a number. "
+            "You can coerce to NaT by passing errors='coerce'"
+        )
+        with pytest.raises(ValueError, match=expected_msg):
+            pd.to_timedelta(invalid_data, errors="raise")
+
     def test_to_timedelta_via_apply(self):
         # GH 5458
         expected = Series([np.timedelta64(1, "s")])