diff --git a/dask/dataframe/shuffle.py b/dask/dataframe/shuffle.py
index 7b229a06666..0c359e2ee24 100644
--- a/dask/dataframe/shuffle.py
+++ b/dask/dataframe/shuffle.py
@@ -1029,10 +1029,12 @@ def fix_overlap(ddf, mins, maxes, lens):
         # this partition (i) if the data from this partition will need to be moved
         # to the next partition (i+1) anyway.  If we concatenate data too early,
         # we may lose rows (https://github.com/dask/dask/issues/6972).
-        if i == len(mins) - 2 or divisions[i] != divisions[i + 1]:
-            frames.append(ddf_keys[i])
-            dsk[(name, i)] = (methods.concat, frames)
-            frames = []
+        if divisions[i] == divisions[i + 1] and i + 1 in overlap:
+            continue
+
+        frames.append(ddf_keys[i])
+        dsk[(name, i)] = (methods.concat, frames)
+        frames = []
 
     graph = HighLevelGraph.from_collections(name, dsk, dependencies=[ddf])
     return new_dd_object(graph, name, ddf._meta, divisions)
diff --git a/dask/dataframe/tests/test_multi.py b/dask/dataframe/tests/test_multi.py
index b5b8d26ecf7..bde70f34ecd 100644
--- a/dask/dataframe/tests/test_multi.py
+++ b/dask/dataframe/tests/test_multi.py
@@ -712,6 +712,18 @@ def test_merge_asof_on_left_right(left_col, right_col):
     assert_eq(result_df, result_dd, check_index=False)
 
 
+def test_merge_asof_with_various_npartitions():
+    # https://github.com/dask/dask/issues/8999
+    df = pd.DataFrame(dict(ts=[pd.to_datetime("1-1-2020")] * 3, foo=[1, 2, 3]))
+    expected = pd.merge_asof(left=df, right=df, on="ts")
+
+    for npartitions in range(1, 5):
+        ddf = dd.from_pandas(df, npartitions=npartitions)
+
+        result = dd.merge_asof(left=ddf, right=ddf, on="ts")
+        assert_eq(expected, result)
+
+
 @pytest.mark.parametrize("join", ["inner", "outer"])
 def test_indexed_concat(join):
     A = pd.DataFrame(
diff --git a/dask/dataframe/tests/test_shuffle.py b/dask/dataframe/tests/test_shuffle.py
index 0c5e75cb104..1b2e53159e4 100644
--- a/dask/dataframe/tests/test_shuffle.py
+++ b/dask/dataframe/tests/test_shuffle.py
@@ -1182,6 +1182,17 @@ def test_set_index_overlap_2():
     assert ddf2.npartitions == 8
 
 
+def test_set_index_overlap_does_not_drop_rows_when_divisions_overlap():
+    # https://github.com/dask/dask/issues/9339
+    df = pd.DataFrame({"ts": [1, 1, 2, 2, 3, 3, 3, 3], "value": "abc"})
+    ddf = dd.from_pandas(df, npartitions=3)
+
+    expected = df.set_index("ts")
+    actual = ddf.set_index("ts", sorted=True)
+
+    assert_eq(expected, actual)
+
+
 def test_compute_current_divisions_nan_partition():
     # Compute divisions 1 null partition
     a = d[d.a > 3].sort_values("a")