There are many species in nature that are fully capable of interbreedig but they do not do so. For example, many ducks have that capability but hybrids are rare in nature. The same is true of many other kinds of animals. The reason is that they have evolved ways to make sure they don't cross species boundaries when they look for mates. Different ducks have different plumages, and the females only respond to courtship by males of their own kind. Since no one is around to police them, you may wonder why they behave themselves. The reason is simple. Each species is adapted to a different way of life. Mallards, for example, feed mainly on plants and they dabble for their food. Another species of duck, such as the surf scoter, dive into deep water looking for clams. To do what they do best, their bodies are build differently. A hybrid in most cases is somewhat intermediate, so it may not dabble as well as a mallard but at the same time it may not be able to dive for clams like the surf scoter. As a result, even if a hybrid can be produced, it may not live very long if it has to compete with mallards and surf scoters. Therefore 2 individuals that are not careful about whom they mate with may wind up not leaving any descendants. Natural selection therefore tends to reward only those individuals that are careful and mate only with their own species. To help them do so, many species have evolved elaborate species identification colors, courtship dances, mating calls, phermones, and so forth, even if they are closely related.

Therefore even if Neanderthals were able to interbreed with humans, if the 2 species remain distinct instead of mixing freely, then they are still not considered the same species. You are correct that there are recent published claims of interbreeding between Neanderthals and humans, but there is no scientific consensus that there was indeed interbreeding. Neanderthal remains have been handled extensively, most often by Europeans, so it could be contamination instead of interbreeding, especially since the data generally shows Europeans to share the most genes with Neanderthals. In any case, these researchers do not claim that humans interbreed freely with Neanderthals. There is for example no data that shows any human female has mitochondrial DNA that belongs to Neanderthals, and also no Y chromsome data to suggest that any European had a Neanderthal father. Until such evidence shows up, there is no evidence that Neandethals actually interbred with modern humans.

it depends on the difference in DNA and if they are in fact a different species or just another branch of the same species i.e. the way different dogs can mate to produce fertile offspring , so going back to the neanderthal we both originated from the same ancestor before so the dna would therefore be similar enough to produce fertile offspring back when they mated

Sigh. I wish people would get over the genetic species definition. Just because it worked for Dobzhansky's fruit flies, he thought that it was a universal rule. It has two major faults. First, it can't be used for 99+% of organisms, the ones that are extinct and the ones that reproduce asexually. Second, There are fishes that not only breed across species lines, they are completely interfertile across generic lines.

this is a definition thing. if two groups of life can breed and produce offspring that themselves can continue to reproduce, the two groups are by definition the same species.

hybrids are everywhere. u don't hv to b the same species to make offspring. same with animals, same with plants

Or, it means that we and Neanderthals were different subspecies, not different species.

I was taught in school that if 2 organisms are genetically similar enough to produce fertile offspring, then they must be the same species. That argument was supported by the fact that hybrids, like mules who were sired by a donkey and a horse are sterile and cannot reproduce. Recently it was discovered that all non-African humans have 1% to 5% neanderthal DNA acquired through inter-species breeding. Doesn't that fact disprove the theory that 2 different species cannot beget fertile offspring when they mate? If so, it opens the door to question settled and solved classifications of modern humans