There are three major high-altitude regions inhabited by humans: highland Ethiopia, Tibet, and the Andean altiplano. In each of these three cases, the locals have adapted in various ways to high altitude – physiological adaptations, as well as cultural. To make it even clearer, those physiological changes are, to a large extent, a consequence of natural selection, rather than individual acclimatization.
There are several recent papers on the genetic basis of human adaptation to high altitude-, and they clearly show that natural selection has taken a different path in each of these populations. This was obvious even before the sequencing started: Andean Amerindians are barrel-chested and have high levels of hemoglobin, while Tibetans pant at the drop of a hat but don’t have particularly high levels of hemoglobin. Moreover, the Tibetan adaptive response works better. Very few Tibetans get chronic mountain sickness, but a significant fraction of Andean highlander do, especially with increasing age. Tibetan babies are significantly plumper than Andean babies. There are hints that the Ethiopian pattern may also be more effective than the Andean, but that hasn’t been studied as much.
Overall, as Cynthia Beall has pointed out, the Tibetan pattern is closer to that seen in animal species that have lived at high altitude for long periods of time.
Rasmus Nielsen concluded the changes in Tibetans happened over about 3,000 years. I doubt that. People have probably been in the altiplano longer than that, and yet their adaptations are substantially less effective.
Hominids have lived in or near Ethiopia and Tibet for much, much longer than anyone has lived in the New World – something like 100 times longer. It is possible that the populations of modern humans living in those areas picked up some altitude-friendly alleles from archaic humans that had lived at high altitude for a very long time – and naturally had more effective adaptations. It only takes a tiny bit of admixture to transmit beneficial alleles. We now have some probable examples of such transmission: it looks as if some archaic HLA alleles (Neanderthal and Denisovan) have reached high frequency in Eurasians, while a very divergent Denisovan version of OAS1, a innate immune gene, is common in Melanesians.
We can be sure that the Andean Indians did not have this opportunity, since humans have only been in the New World for 15,000 years or so. No llama jokes, please.