Bred their way into us?? No way. Studies of neanderthal DNA indicate that it was about halfway between ours and that of a chimpanzee, clearly explaining why there was never any evidence of interbreeding. We could no more interbreed with them than we could with horses and, further, all scientists agree, there is no way we are descended from them.
That presents yet another insoluble problem for evolutionists: to believe that modern man evolved, there would have to be some closer hominid, i.e. some hominid between us and the neanderthal which we COULD have descended from and, since this closer hominid would be closer to us both in time and morphology than the neanderthal, his works and remains should be very easy to find; they should be all over the place. Neanderthal works and remains are all over the place. Neanderthal was the major inhabitant of this planet just prior to us. Here's one of Jay Matternes' modern reconstructions(not based on diseased specimens) of what they looked like:
Not really all that bad looking. Not one of us, but a thoroughly modern man despite the DNA difference.