And I would say the answer's yes.
You could make it so that the cluster centers sit right on top of each other,
and the separation line looks like this.
And all the top points are associated to the top cluster center,
and all the bottom points are associated to the bottom cluster center.
Granted, it's unlikely, to have init-,
,initialization like this, but
if it happens, then the algorithm would believe this is one cluster.
And this is another cluster.
If we re-run it and you initialize differently.
Say one of the cluster centers sits over here.
Then a separation line will fall like that.
And the classes would automatically resolve themselves.
It's unlikely, but there exists a bad local minimum,
even in the example I showed you over here.
Now as a rule of thumb,
the more cluster centers you have, the more local minima you find.
But they exist, as a result, you are forced to run the algorithm multiple times.