2 from Belgium and Gibralter from before "Neanderthals" were identified as not being H. sapiens(from here)
7 from the Neanderthal "type site" in Germany
2 from Grotto of Spy d'Orneau in Belgium
between 24 - 36 individuals from Croatia (remains are fragmentary)
1 near-complete skeleton from La-Chapelle-aux-Saints in France
1 from Saint-Cesaire in France
1 from Arcy-sur-Cure, France
6 from the Moula-Guercy cave, above the modern Rhone River, France
9 from the Shanidar cave in Iraq
*at least* 13 from sites at Tabun, Qafzeh, and Skhul near Mount Carmel, Israel
...there were more paper references turned up in my search, but I did not track them all down.
While exact numbers don't seem to be available from this, simple summing of the ones above yield a minimum number of 66 individuals. I remember seeing somewhere that we actually have remains of almost 200 Neanderthal individuals, but I cannot substantiate that at the moment.
And while this:
Neanderthal fossils have previously been found over a large area, stretching from the Mediterranean as far east as Uzbekistan. Most remains are fragmentary, however, so it can be difficult to determine whether a fossil is of Neanderthal or of modern human origin.
(from here) doesn't actually mention specific numbers, it does imply a few more than 5. It does mention two specific sets of fossils, one of a child from Teshik-Tash in Uzbekistan, and one set of adult fossils from Okladnikov in southern Siberia.
And this article says "Thousands of Neanderthal fossils and artifacts are known", though does not give any specific list.
This one gives: "In 1887, two complete skeletons were found in a cave near Spy in Belgium, and more from sites in France in 1887, 1908 and 1911." which seems to match a few from the first quote.
Talk Origins lists around five 'prominent' fossils, again, most listed in the first link. "...there are now thousands of hominid fossils. They are however mostly fragmentary, often consisting of single bones or isolated teeth. Complete skulls and skeletons are rare."
Neanderthal skeletons have been found in caves and shelters across Europe, in southwest Asia, and eastward to Uzbekistan in Central Asia, providing abundant skeletal remains and associated archaeological material for understanding these prehistoric humans. The Neanderthals are now known from several hundred individuals, represented by remains varying from isolated teeth to virtually complete skeletons.And here's a timeline. Note that not all entries are Neanderthal finds, but the majority appear to be.
Apologetics press points out that there have been mis-datings made, though, and I haven't gone in to check all the dates of the articles above against the date of the 'dating disaster', but I think they might be overreacting to one incident against much other information. As does Panda's Thumb.
...and as for the arthritis comment, the following is from a review of the book "Buried Alive: The Startling Truth about Neanderthal Man" by Jack Cuozzo, who is a creationist.
In four and a half pages (pp.274-279) he [Cuozzo] demolishes the notion that the distinctive Neandertal morphology is entirely due to disease, taking apart the three proposed hypotheses - arthritis, syphilis, rickets - one by one; he even chastises a fellow creationist, Lubenow, for getting caught up in the rickets hypothesis.