The causes of disappearance of the Neanderthals, the only human population living in Europe before the arrival of Homo sapiens, have been debated for decades by the scientific community. Different hypotheses have been advanced to explain this demise, such as cognitive, adaptive and cultural inferiority of Neanderthals. Here, we investigate the disappearance of Neanderthals by examining the extent of demographic changes needed over a period of 10,000 years (yrs) to lead to their extinction. In regard to such fossil populations, we inferred demographic parameters from present day and past hunter-gatherer populations, and from bio-anthropological rules. We used demographic modeling and simulations to identify the set of plausible demographic parameters of the Neanderthal population compatible with the observed dynamics, and to explore the circumstances under which they might have led to the disappearance of Neanderthals. A slight (<4%) but continuous decrease in the fertility rate of younger Neanderthal women could have had a significant impact on these dynamics, and could have precipitated their demise. Our results open the way to non-catastrophic events as plausible explanations for Neanderthal extinction.