[According to] Pekka Hämäläinen’s Comanche Empire, the Comanche maintained about 20,000 lives on a herd of 5,000,000 buffalo, but were in a constant state of economic crisis. They required trade with agriculturalists for beans and corn to keep their children healthy and had to make war on agriculturalists constantly to provide other resources.
One minor thread in Hämäläinen’s book, perking along in the background, is the fragility of all the hunter-gatherer and quasi-agrictultural societies between the Missisippi and the Rockies. A single epidemic, or drought, or defeat in battle could shatter their communities. Their survivors would be consumed by their neighbors and disappear from history, save for some vague reference in the letters of a European traveler or missionary who’d heard of their fate through gossip on the trade networks.
From here (comment 56) (a webpage for the book cited can be found at the site of its publisher).
There is evidence that the Comanche hunting lifestyle looked a lot like that of the Neanderthals:
We report here on the direct isotopic evidence for Neanderthal and early modern human diets in Europe. Isotopic methods indicate the sources of dietary protein over many years of life, and show that Neanderthals had a similar diet through time (≈120,000 to ≈37,000 cal BP) and in different regions of Europe. The isotopic evidence indicates that in all cases Neanderthals were top-level carnivores and obtained all, or most, of their dietary protein from large herbivores. In contrast, early modern humans (≈40,000 to ≈27,000 cal BP) exhibited a wider range of isotopic values, and a number of individuals had evidence for the consumption of aquatic (marine and freshwater) resources. This pattern includes Oase 1, the oldest directly dated modern human in Europe (≈40,000 cal BP) with the highest nitrogen isotope value of all of the humans studied, likely because of freshwater fish consumption.
These two facts, make the following estimate of the size of the Neanderthal population, based upon the amount of genetic diversity in a handful of available Neanderthal DNA sequences, seem plausible:
[G]enetic evidence from the remains of six Neanderthals (Homo neanderthalensis) suggests the population hovered at an average of 1,500 females of reproductive age in Europe between 38,000 and 70,000 years ago, with the maximum estimate of 3,500 such female Neanderthals.
If the genetic evidence based estimate is correct, the population of Neanderthals in Europe was 3,000-7,000 individuals, and at times less, in the approximately time frame that modern humans arrived on the scene. Even if this estimate has a great margin of error because the sample size is so small, it provides an order of magnitude estimate.
This isn't out of line with the Comanche figure of 20,000 for essentially the entire Southwestern United States. The Comanche should be greater than a comparable population for Neaderthals because they had technological advantages (horses, guns), a more mild climate for much of the time period, and inhabitable Europe was smaller in the time of the Neanderthals than it is now. The Comanches in the 1700s and 1800s, unlike the Neanderthals or pre-Columbia big game hunters in the New World, had at their disposal horses (about 120,000 of them) and various trade goods including guns and other metal instruments that they did not manufacture themselves. The Neanderthals had only crude stone and wood tools and did not even have domesticated dogs. Also, in the Neanderthal era, a good share of Europe was too frigid for Neanderthals or the animals they hunted to survive in since it was under a giant ice sheet.
Another study I've seen (I've misplaced the link for the moment) suggests that the Neanderthal population may have been further divided into three regional subgroups (roughly speaking, Western, Southern and Eastern). I've also seen estimates that hunter-gatherer groups typically consisted of a few dozen people. This suggests that each of the three regional subgroups of Neanderthals may have consisted of around thirty to sixty small tribal groups.
Another interesting reference number is that the peak population of Siberian Tigers, whose ecological niche was as a top level carnivore in a place somewhat similar in ecology and size to the Europe that Neanderthals inhabited is estimate to be about 2000, which is on the same order of magnitude as the estimated pre-modern human Neanderthal population. Siberia is also a nice comparison because it has been so thinly populated by humans.
There numbers are much lower than estimates I've seen for the total pre-Columbian population of North American placed at 10 million, but many pre-Columbian Native Americans weren't trying to provide the vast majority of their diet from hunting large herbivores, as it appears that the Neanderthals did. The indigenous population of Australia prior to 1788 first contact, is estimated at about 315,000. At that point, however, Australian megafauna that were present when humans first arrived on the continent, had long since gone extinct, so a population estimate based on this hunter-gatherer society may not be comparable.
The notion that contact with modern humans could in some way wipe out an ecologically fragile population of less than seven thousand individuals spread across Europe surviving almost exclusively by hunting big game is far more plausible than it would be if the population of Neanderthals had been in the hundreds of thousands or millions at the time. The multiple cause downfall of the Comanche empire described in the book cited above offers one model of what the demise of the Neanderthal might have looked like after first contact with modern humans.