There were still large pachyderms (either mastodons or mammoths) in Florida as recently as 5000 years ago. It is pretty much accepted they were the victims of Neolithic hunters. It is possible for subsistence hunters to wipe out species, such as many of North America's large mammals, or even to change entire ecosystems (sometimes for the better!). The aboriginal Australians, through controlled burning, were able to greatly increase the productivity of vast sections of the Outback by keeping wildfires down and by clearing brush and allowing fresh growth. This led to species diversity and more food.

As for the Great Plains Buffalo Culture, we should keep in mind we intervened in that experiment only two brief centuries after the introduction of the horse. It was really just getting started when we showed up and spoiled the party. If the palefaces had shown up a century later, I think things might have looked very different. We might have had Dakota and Cheyenne war parties raiding the Eastern US the way the Asian hordes harried medieval Europe.

Many "primitive" cultures, such as the Balinese, the Andean and Yucatan civilizations, even the ancient Egyptians, were able to institute stable, long-lived and complex agricultural systems that were extremely productive and able to exploit the harsh natural environments they inhabited. They supported massive populations. In other cases, overgrazing or overlogging either made marginal areas uninhabitable, or vulnerable to natural change (Chaco Canyon comes to mind). Still, as a general rule, I think it's safe to say that human activities, even organized into hydraulic civilizations like in China and Mesopotamia, usually evolve into some kind of relatively stable relationship with their natural environment, given enough time.

But when man graduated from muscle/water/wind power to fossil fuels, the world changed.
We have much more power and knowledge to better deal with the environment now, but for social and economic reasons, we usually move to fast to learn the lessons our Stone and Bronze Age ancestors mastered.