diff --git a/pandas/core/dtypes/dtypes.py b/pandas/core/dtypes/dtypes.py
index aa7e6801ba431..fcdb89dd8a334 100644
--- a/pandas/core/dtypes/dtypes.py
+++ b/pandas/core/dtypes/dtypes.py
@@ -685,7 +685,7 @@ def __init__(self, unit="ns", tz=None):
             tz = timezones.tz_standardize(tz)
         elif tz is not None:
             raise pytz.UnknownTimeZoneError(tz)
-        elif tz is None:
+        if tz is None:
             raise TypeError("A 'tz' is required.")
 
         self._unit = unit
@@ -737,14 +737,17 @@ def construct_from_string(cls, string):
         """
         if isinstance(string, str):
             msg = "Could not construct DatetimeTZDtype from '{}'"
-            try:
-                match = cls._match.match(string)
-                if match:
-                    d = match.groupdict()
+            match = cls._match.match(string)
+            if match:
+                d = match.groupdict()
+                try:
                     return cls(unit=d["unit"], tz=d["tz"])
-            except Exception:
-                # TODO(py3): Change this pass to `raise TypeError(msg) from e`
-                pass
+                except (KeyError, TypeError, ValueError) as err:
+                    # KeyError if maybe_get_tz tries and fails to get a
+                    #  pytz timezone (actually pytz.UnknownTimeZoneError).
+                    # TypeError if we pass a nonsense tz;
+                    # ValueError if we pass a unit other than "ns"
+                    raise TypeError(msg.format(string)) from err
             raise TypeError(msg.format(string))
 
         raise TypeError("Could not construct DatetimeTZDtype")
diff --git a/pandas/tests/dtypes/test_dtypes.py b/pandas/tests/dtypes/test_dtypes.py
index 3288c9c584565..6013e58bc6929 100644
--- a/pandas/tests/dtypes/test_dtypes.py
+++ b/pandas/tests/dtypes/test_dtypes.py
@@ -248,9 +248,19 @@ def test_construct_from_string_raises(self):
         with pytest.raises(TypeError, match="notatz"):
             DatetimeTZDtype.construct_from_string("datetime64[ns, notatz]")
 
-        with pytest.raises(TypeError, match="^Could not construct DatetimeTZDtype$"):
+        msg = "^Could not construct DatetimeTZDtype"
+        with pytest.raises(TypeError, match=msg):
+            # list instead of string
             DatetimeTZDtype.construct_from_string(["datetime64[ns, notatz]"])
 
+        with pytest.raises(TypeError, match=msg):
+            # non-nano unit
+            DatetimeTZDtype.construct_from_string("datetime64[ps, UTC]")
+
+        with pytest.raises(TypeError, match=msg):
+            # dateutil str that returns None from gettz
+            DatetimeTZDtype.construct_from_string("datetime64[ns, dateutil/invalid]")
+
     def test_is_dtype(self):
         assert not DatetimeTZDtype.is_dtype(None)
         assert DatetimeTZDtype.is_dtype(self.dtype)