In fact there was always a big mystery as to why there was zero evidence of crossbreeding despite the two groups living in close proximity for long periods of time. The DNA studies pretty much resolved the mystery. WE could no more interbreed with neanderthals than we could with horses. Moreover, doing it with a neanderthal would not be that much different than doing it with an ape.
There is much reason to think that modern humans could not interbreed with Neanderthals, simply because the vast preponderence of the evidence points squarely toward no interbreeding. But that being said, even if modern humans and Neanderthals could interbreed there are many, many sociocultural and behavioral factors that could very well make it never happen, or happen so rarely as to be of no consequence.
In short, for all we know Neanderthals may've been inclined to kill modern humans on sight, or vice versa. There's certainly plenty of fossil evidence to establish that Neanderthal fossils vanish from any given region almost immediately after the arrival of Cro-Magnons.
We know that over and over again that tends to be the initial reaction of aboriginal humans, for that matter. It isn't 'Oh, come screw us'; it's 'Get the $%#& off our land!" Eventually, by conquest or by diplomacy, a rapport gets established, and then the screwing comes along, but to reiterate yet again, there is no reason to think that we could communicate with Neanderthal in anything but the most rudimentary fashion.
Would you want to go to sleep beside some incomprehensible wild-eyed burly ape-woman scheming how to bash your head in with a rock for all you know?
The DNA studies pretty much resolved the mystery.There have been a couple of mtDNA studies; the first one used fewer than 400 base pairs -- assumed to be from the ancient sample and not from, say, bacteria -- out of a presumed 16,000+, and used that data to claim, rather grandiosely, that the divergence was too great. It was GIGO. And mtDNA studies can't tell whether two people can interbreed, because mtDNA has nothing to do with interbreeding.