Interbreeding between H. sapiens and neandertalis was quite limited.
Our results indicate that the amount of Neanderthal DNA in living non-Africans can be explained with maximum probability by the exchange of a single pair of individuals between the subpopulations at each 77 generations, but larger exchange frequencies are also allowed with sizeable probability.
--Extremely Rare Interbreeding Events Can Explain Neanderthal DNA in Living Humans
... the interbreeding success rate between humans and Neanderthals was found to be below 2% (Fig. 2, Table 1, Fig. S1, and Table S1). Under demographic scenario A (Table 1), which is based on the most realistic demographic parameters, we estimate this interbreeding success to be even well below 1% (0.51%, mode of black curve in Fig. 2; 95% confidence interval: 0.33–0.89%). ... If we assume that both species interacted for 10,000 y during the range expansion of modern humans (11), successful admixtures between Neanderthals and humans would have occurred, on average, only once every 23–50 y over the whole hybrid zone to produce introgression levels of 1–3%, which shows that they were extremely rare events.
--Strong reproductive isolation between humans and Neanderthals inferred from observed patterns of introgression
Note that this technically doesn't touch on "interbreeding" but rather addresses successful interbreeding; sterile individuals won't be detected in this approach, and even sub-fertile individuals (whose lineages died out) will be underrepresented.
We have strong evidence that male progeny of sapiens/neandertalis crosses were either sterile or extremely sub-fertile:
These results suggest that part of the reduction in Neandertal ancestry near genes is due to Neandertal alleles that reduced fertility in males when moved to a modern human genetic background.
--The landscape of Neandertal ancestry in present-day humans
And even apart from that, there's good evidence that hybrids were less "fit" (in evolutionary terms) than either parent:
After hybridization, on average, selection appears to have removed Neanderthal alleles from the human population. ... We find that the bulk of purifying selection against Neanderthal ancestry is best understood as acting on many weakly deleterious alleles. We propose that the majority of these alleles were effectively neutral-and segregating at high frequency-in Neanderthals, but became selected against after entering human populations of much larger effective size. While individually of small effect, these alleles potentially imposed a heavy genetic load on the early-generation human-Neanderthal hybrids.
--The Strength of Selection against Neanderthal Introgression
So it's very likely that the rate of interbreeding, per se, must have been higher than the roughly once-per-generation rate that can be inferred from genome analysis, but most of the hybrids were sterile, or poorly adapted. Of course, this is exactly what you'd expect from interspecies hybridization; the number of successful sapiens/neandertal hybrids through the millennia of their interactions was probably lower than the number of fertile mules in the past few hundred years.
