But they lived together in overlapping territory for tens of thousands of years. I don't know about your idea on this one? (maybe?)
The territory shared seems to be at the very Southern end of the Neanderthal's range. There seems to have been a steady migration of Cro-Magnon from the South, IMHO it would be only a matter of time when some migrants from Africa brought a bug that the Neanderthals could not handle.
The fact that the ranges overlapped for some 10,000 years and then Neanderthal disappeared in all locations in a short period of time points to some type of catastrophe befalling the Neanderthals but not the Cro-Magnon.