diff --git a/doc/source/whatsnew/v0.24.0.txt b/doc/source/whatsnew/v0.24.0.txt
index 68c1839221508..43c75cde74b42 100644
--- a/doc/source/whatsnew/v0.24.0.txt
+++ b/doc/source/whatsnew/v0.24.0.txt
@@ -24,6 +24,41 @@ Other Enhancements
 Backwards incompatible API changes
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
+.. _whatsnew_0240.api.datetimelike.normalize
+
+Tick DateOffset Normalize Restrictions
+--------------------------------------
+
+Creating a ``Tick`` object (:class:``Day``, :class:``Hour``, :class:``Minute``,
+:class:``Second``, :class:``Milli``, :class:``Micro``, :class:``Nano``) with
+`normalize=True` is no longer supported.  This prevents unexpected behavior
+where addition could fail to be monotone or associative.  (:issue:`21427`)
+
+.. ipython:: python
+
+    ts = pd.Timestamp('2018-06-11 18:01:14')
+    ts
+    tic = pd.offsets.Hour(n=2, normalize=True)
+    tic
+
+Previous Behavior:
+
+.. code-block:: ipython
+
+    In [4]: ts + tic
+    Out [4]: Timestamp('2018-06-11 00:00:00')
+
+    In [5]: ts + tic + tic + tic == ts + (tic + tic + tic)
+    Out [5]: False
+
+Current Behavior:
+
+.. ipython:: python
+
+    tic = pd.offsets.Hour(n=2)
+    ts + tic + tic + tic == ts + (tic + tic + tic)
+
+
 .. _whatsnew_0240.api.datetimelike:
 
 Datetimelike API Changes
diff --git a/pandas/tests/tseries/offsets/test_offsets.py b/pandas/tests/tseries/offsets/test_offsets.py
index 8bf0d9f915d04..33e5a70c4c30b 100644
--- a/pandas/tests/tseries/offsets/test_offsets.py
+++ b/pandas/tests/tseries/offsets/test_offsets.py
@@ -28,7 +28,7 @@
                                     YearEnd, Day,
                                     QuarterEnd, BusinessMonthEnd, FY5253,
                                     Nano, Easter, FY5253Quarter,
-                                    LastWeekOfMonth)
+                                    LastWeekOfMonth, Tick)
 from pandas.core.tools.datetimes import format, ole2datetime
 import pandas.tseries.offsets as offsets
 from pandas.io.pickle import read_pickle
@@ -270,6 +270,11 @@ def test_offset_freqstr(self, offset_types):
 
     def _check_offsetfunc_works(self, offset, funcname, dt, expected,
                                 normalize=False):
+
+        if normalize and issubclass(offset, Tick):
+            # normalize=True disallowed for Tick subclasses GH#21427
+            return
+
         offset_s = self._get_offset(offset, normalize=normalize)
         func = getattr(offset_s, funcname)
 
@@ -458,6 +463,9 @@ def test_onOffset(self, offset_types):
         assert offset_s.onOffset(dt)
 
         # when normalize=True, onOffset checks time is 00:00:00
+        if issubclass(offset_types, Tick):
+            # normalize=True disallowed for Tick subclasses GH#21427
+            return
         offset_n = self._get_offset(offset_types, normalize=True)
         assert not offset_n.onOffset(dt)
 
@@ -485,7 +493,9 @@ def test_add(self, offset_types, tz):
         assert isinstance(result, Timestamp)
         assert result == expected_localize
 
-        # normalize=True
+        # normalize=True, disallowed for Tick subclasses GH#21427
+        if issubclass(offset_types, Tick):
+            return
         offset_s = self._get_offset(offset_types, normalize=True)
         expected = Timestamp(expected.date())
 
@@ -3098,6 +3108,14 @@ def test_require_integers(offset_types):
         cls(n=1.5)
 
 
+def test_tick_normalize_raises(tick_classes):
+    # check that trying to create a Tick object with normalize=True raises
+    # GH#21427
+    cls = tick_classes
+    with pytest.raises(ValueError):
+        cls(n=3, normalize=True)
+
+
 def test_weeks_onoffset():
     # GH#18510 Week with weekday = None, normalize = False should always
     # be onOffset
diff --git a/pandas/tseries/offsets.py b/pandas/tseries/offsets.py
index a5a983bf94bb8..ecd15bc7b04b8 100644
--- a/pandas/tseries/offsets.py
+++ b/pandas/tseries/offsets.py
@@ -2217,8 +2217,10 @@ class Tick(SingleConstructorOffset):
     _attributes = frozenset(['n', 'normalize'])
 
     def __init__(self, n=1, normalize=False):
-        # TODO: do Tick classes with normalize=True make sense?
         self.n = self._validate_n(n)
+        if normalize:
+            raise ValueError("Tick offset with `normalize=True` are not "
+                             "allowed.")  # GH#21427
         self.normalize = normalize
 
     __gt__ = _tick_comp(operator.gt)