With ground up bone fragments - about the mass of a pill - taken from 3 Neanderthal females whose remains were recovered at the Vindija site in Croatia, more has been learned through genetically sequencing Neanderthals than from studying them in the fossil record. Historically to science, Neanderthals stretch back to 1856 when the Feldhofer Grotte yielded the first Neanderthal specimen, a skull-cap and some bones.
The bones from the Vindija cave fall within the range of 38-44Kyr. Neanderthal DNA appears to be packed into the human genome randomly, suggesting that it did not confer survival advantages on modern humans.  Interestingly, no evidence for gene flow was found, moving in the opposite direction - from Homo Sapiens to Neanderthals. This fact might be accounted for if the interspecific, "love-fest" in Europe occurred between small, invading wavelets of modern humans and a large, settled population of Neanderthals (or between a wave of modern humans whose population-size was expanding and a resident population of Neanderthals whose population-size was contracting). The lack of human genes in the Vindija females' bones could - simply - be an artifact of their times; Modern humans may not have arrived in their vicinity 40Kyr ago.
Subsequent to the dispersal of Homo Sapiens from Africa, the evidence is that Homo Sapiens and Homo Neanderthalensis interbred. In present-day non-African population groups, a 1-4% contribution of Neanderthal genes has been identified. In the present-day African population, there is an estimated, statistically significant non-zero percent of Neanderthal genes, indicating small, intermittent flows of Neanderthals (or back-flows of hybridized humans) from Western Asia into Africa. (There has never been a convincing Neanderthal site unearthed in Africa.)
For decades, tantalizing clues pointing towards assimilation have been put forth, yet nothing compelling came about to cinch the case - until relatively recently. Some calculations ran that it would have taken only one absorption of a Neanderthal individual every other generation into the Homo Sapien population (over a period of about 10Kyr) for Neanderthal genes to become detectable in current human populations.
In 1997, Wolpoff and Caspari identified Neanderthal, non-adaptive traits in Europeans that cannot easily be reconciled with OoA. Wolpoff's and Caspari's evidence was dismissed or deemed "fragmentary". This evidence that humans and Neanderthals exchanged genes in Europe was blamed - not the OoA theory itself, which excludes the possibility of interbreeding.  In 2002, Eswaran described what some have termed "non-bottle neck traits" in humans.  According to OoA, the entire human population under-went a constriction after leaving Africa - squeezing its genetic variation down to the aperature of a capillary - which rules out "non-bottle neck traits" being found in living human populations. The Human Genome Project, a great scientific achievement, linked no more than 90% of existing human mtDNA and Y haplo-groups as deriving from the African population that spread out 70-100Kyr ago, revealing another possible discrepancy between solid, genetic evidence and the OoA hypothesis.
I'll not rehearse anymore older, confounding evidence against OoA here, but suffice it to say that its demise was smelt drifting palpably in the air for some time, and in retrospect why it garnered such a long-term, almost fanatical adherence might best be explained that it is an idea, resounding with a politicized knell of "truth". Over time, if possible, when counter-evidence against OoA mounted, its acolytes became more and more "hardened" towards its invincibility. 
On the face of it and from the outset (1985), the OoA hypothesis was uniquely susceptible to refutation; It is a greedy explainer, plus it is constructed on top of a universal negation. Under it, NO present-living human can trace any of his ancestry to a population other than the one which radiated from Africa 70-100Kyr ago. Any detectable amount of interbreeding between modern humans and Neanderthals blows this universal negation to shreds. The analysis of the Vindija bones blew it up. Simply put, OoA cannot countenance evidence of assimilation - surviving intact.
John Hawks expressed it best:
"From now on, we are all multiregionalists trying to explain the out-of-Africa pattern."
1. In paleoanthropological argot, the phrase for the OoA hypothesis is the "Recent Single-Origin Hypothesis". "Single-Origin" refers to the exclusive African origin of modern humans. Evidence of interbreeding (found in current human populations) between modern humans and any other hominid species on one other continent yields a "Recent Dual-Origin Hypothesis".
2. However in Science August 25, 2011: as a result of our ancestors' interbreeding with more than one non-human species, the immune systems of modern humans were "invigorated". Two extinct species (Neanderthals and Denisovans) were shown to have contributed HLA variants to modern human populations. These HLA complexes "strengthened" the capacity of the human immune system to resist infections (and increased the "viability" of the human genome). This study refutes a once widely-held insistence that interbreeding with archaics conveyed NO survival advantages on humans. DNA studies of Neandertals and Denisovans suggest that these two species - together - account for ~7.5% of the ancestry of modern Melanesians. Denisovan FAQ
3. Wolpoff, M. and Caspari, R. (1997) from Race and Human Evolution, A Fatal Attraction:
4. Vinayak Eswaran (2002) from: A Diffusion Wave out of Africa: The Mechanism of the Modern Human Revolution?:
"There is also regional continuity in nonadaptive features. Nonadaptive features can persist after they are established at high frequency, when no evolutionary forces act to change them significantly. Because of their nature, they are very unlikely to persist if the history of a region is marked by population replacements. One of the best arguments for a significant European Neandertal input into the gene pools of later Europeans is the persistence of nonadaptive traits. Features such as the shape of the hole for the mandibular nerve to enter the mandible, or the presence of a small bump on the outer edge of the femur shaft, near its top, are difficult to explain any other way."
"Not all genetic studies show signs of bottlenecks. For example, proteins, blood groups, and alleles of the major histocompatibility complex (MHC) loci show non-Africans having a genetic diversity comparable to and often greater than that of Africans (Takahata 1995). The presence of numerous ancient alleles at the MHC loci has been interpreted as suggesting that no population bottlenecks may ever have occurred in human and hominoid evolution (Ayala 1995). Other genetic systems too do not carry the bottleneck-and-expansion signature seen in mtDNA and at some other loci. These contrasting indications from different genetic studies need an explanation, particularly because the presence of bottlenecks has been construed as supporting a replacement scenario for the modern transition. I will now show that the dichotomy in the genetic data strongly suggests that assimilation from archaic populations occurred during the modern transition."
"Assimilation explains why proteins, blood groups, MHC loci, etc., show no signs of an Out-of-Africa bottleneck. Thus, the empirical fact of the absence of bottlenecking in ancient polymorphism implies that assimilation, not replacement, is the best explanation of the genetic data."
"In Europe, long thought to be the best source of evidence for replacement, the fossil record offers equally little comfort to Eve's supporters. The evidence points to much mixing between the invaders and the native Neanderthals. According to our own analysis, many features once thought unique to Neanderthals lingered on among later Europeans. Only a few Neanderthal features disappear completely from the fossil record with the demise of the group as a whole."