As far as I know there is currenly little doubt about exchanging some DNA fragments between H. sapiens and H. Neanderthalensis. The original paper of Green et al. (Science 328 (2010) p. 710) explains in detail the reasons why the observed similarities in some DNA segments are admixtures rather than effects of other events. They estimated the Neanderthal admixture in modern, non-African human DNA to be ca. 2%.
Is the Green et al paper available through ResearchGate? It would be nice to be able to have a look at this.
I've not read the paper, but to my view, however cautious the data aquisition is, it's still hard to set apart contamination from admixture.
This is why a few samples from different caves were analyzed. The probability of having the same contamination in different samples is quite low. The team working with Neanderthal DNA is one of the most experienced in this kind of work so they know how to deal with contamination issues and data integrity.
Neanderthals were not so homogeneous as erroneously scientists believe!
Based on the skeletal material there were at least 5 distinctive races (or sub-species) in the so called "Neanderthal humans"!
Even their toolkit was very much differentiated and exhibited a vast variety. If we were able to DNA sequencing two or three individuals of each of the above mentioned sub species we could have been able to have an opinion. But now with only one full ancient genome sequencing of a Neanderthal who RATIONALLY is not a representative of all the Neanderthalian sub-species and racial gradients we can't say anything but only speculate.
In concluding even if Neanderthals were able to produce fertile offspring when mating with Early or Archaic Homo Sapiens that does not mean that this group made it up to day. It is very possible that other races or tribes destroyed them because the Gravettian for example human morphology differs quite a lot from the Aurignacian, even from the Aurignacians that we are sure 100% that they were Homo Sapiens Sapiens.
So the answer is that yes, Neanders and Sapiens could have mixed but even if they did who knows if their offspring was fertile and even if it was, if it survived the following intrusions of the Paleolithic and Mesolithic Eras.
We can find out by doing a lot and most importantly EXPENSIVE research, but to be honest i don't see that coming in the near future.