diff --git a/distributed/diagnostics/graph_layout.py b/distributed/diagnostics/graph_layout.py
index 69ab771450..02565f9027 100644
--- a/distributed/diagnostics/graph_layout.py
+++ b/distributed/diagnostics/graph_layout.py
@@ -15,6 +15,7 @@ class GraphLayout(SchedulerPlugin):
     def __init__(self, scheduler):
         self.x = {}
         self.y = {}
+        self.collision = {}
         self.scheduler = scheduler
         self.index = {}
         self.index_edge = {}
@@ -62,6 +63,14 @@ def update_graph(self, scheduler, dependencies=None, priority=None,
                 y = self.next_y
                 self.next_y += 1
 
+            if (x, y) in self.collision:
+                old_x, old_y = x, y
+                x, y = self.collision[(x, y)]
+                y += 0.1
+                self.collision[old_x, old_y] = (x, y)
+            else:
+                self.collision[(x, y)] = (x, y)
+
             self.x[key] = x
             self.y[key] = y
             self.index[key] = self.next_index
@@ -85,6 +94,11 @@ def transition(self, key, start, finish, *args, **kwargs):
             for dep in task.dependencies:
                 self.visible_edge_updates.append((self.index_edge.pop((dep.key, key)), 'False'))
 
+            try:
+                del self.collision[(self.x[key], self.y[key])]
+            except KeyError:
+                pass
+
             for collection in [self.x, self.y, self.index]:
                 del collection[key]
 
diff --git a/distributed/diagnostics/tests/test_graph_layout.py b/distributed/diagnostics/tests/test_graph_layout.py
index f48168ee3f..63ecb0c700 100644
--- a/distributed/diagnostics/tests/test_graph_layout.py
+++ b/distributed/diagnostics/tests/test_graph_layout.py
@@ -1,3 +1,5 @@
+import operator
+
 from distributed.utils_test import gen_cluster, inc
 from distributed.diagnostics import GraphLayout
 from distributed import wait
@@ -79,3 +81,16 @@ def test_forget(c, s, a, b):
     assert not gl.y
     assert not gl.index
     assert not gl.index_edge
+    assert not gl.collision
+
+
+@gen_cluster(client=True)
+def test_unique_positions(c, s, a, b):
+    gl = GraphLayout(s)
+
+    x = c.submit(inc, 1)
+    ys = [c.submit(operator.add, x, i) for i in range(5)]
+    yield wait(ys)
+
+    y_positions = [(gl.x[k], gl.y[k]) for k in gl.x]
+    assert len(y_positions) == len(set(y_positions))