No humans alive today are descended in any part from Neanderthals. It is believed that Modern Humans and Neanderthals could not breed with each other, because Neanderthals and the great apes had 48 chromosomes whereas modern humans have only 46.
"An international group of German and US scientists successfully extracted tiny DNA fragments from one of the bones of a Neanderthal museum specimen unearthed in 1856. By using a technique known as polymerase chain reaction, they were able to find fragments derived from a specific portion of the mitochondrion DNA, and then to compare the DNA sequences of those fragments to similar regions of modern human DNA. The results, which they carefully controlled to rule out contamination with foreign or modern DNA, were conclusive. Neanderthal mitochondrial DNA was, just as you might expect, much closer to modern human DNA than any other living species, including the chimpanzee. However, the differences between Neanderthal and modern DNA falls well outside the range of variation within the modern human species.
These results show that Neanderthals split off from the line that gave rise to modern humans well before any of the different geographic groups of modern humans (Asian, African, or European) were established. In plain language, this means that Neanderthals were NOT the direct ancestors of any modern humans. In fact, the degree of DNA sequence difference led the investigators to suggest that Neanderthals diverged from modern humans sometime between 550,000 and 690,000 years ago."
References: "Neanderthal DNA sequences and the origin of modern humans" in the journal CELL, volume 90, pages 19-30 (The July 11, 1997 issue). In the same issue of CELL, Tomas Lindahl wrote an excellent review of the work called "Facts and artifacts of ancient DNA." This appears on page 1 of the July 11th issue."
There is no evidence that there was any inter-breeding between Cro-Magnon and Neanderthals.
On page 128 of Bryan Sykes' 2001 book (7 Daughters of Eve), at page 128 he states:
"At some point in six million years since humans and great apes split away from our mutual common ancestor, two chromosomes that are still separate in the great apes fused together in the human lineage to produce our chromosome 2. There is no knowing at which point along out own lineage this chromosome fusion occurred, but if it happened after the split between the lines that became Cro-Magnon and Neanderthal then there would be a chromosome imbalance, with Neanderthals having forty-eight chromosomes and Cro-Magnons only forty-six."