This answer might be more appropriate to /r/askanthropology, but with North American pre-Columbian timelines the line between history and myth is pretty blurry. One of the only sources of Cherokee oral traditions, Mooney's Myths of the Cherokee (1900), is available online. The fourth chapter, "Origin of Disease and Medicine," tells the story of how Little Deer, the chief of the deer people, punishes overhunting with "deer sickness," which is commonly held to be rheumatoid arthritis. The reason why this became necessary is for two reasons (within the story):
a) people (i.e., Cherokee) had started to hunt for fun rather than necessity;
b) and new technology (bows, atlatls, hooks, etc.) gave them a huge advantage over other people (the animals).
This then created a situation where, to quote Mooney (p. 250):
Man invented bows, knives, blowguns, spears, and hooks, and began to slaughter the larger animals, birds, and fishes for their flesh or their skins, while the smaller creatures, such as the frogs and worms, were crushed and trodden upon without thought, out of pure carelessness or contempt.
The solution from the council of animals was two-fold: the Little Deer would cripple the hands of hunters with arthritis so they couldn't draw a bowstring; and hunters willing to practice respectful hunting through ceremony could be cured of their arthritis. This is why the title of the story is "origin of disease and medicine." Disease came into being as a result of overhunting animals, and medicine (through ceremony) was a cure by reminding people of their proper place in the world and proper action.
All of this is a long-winded way of getting around to this: many people consider this story to be an indication that at least some Eastern North American native peoples were aware that overharvest could lead to negative effects on populations of animals. As for the validity of the story (Mooney is questioned for many reasons) as history, there is some corroborating (anecdotal) evidence:
I was taught a related story from my grandfather in a different language as a part of learning to hunt as a child (he was Shawnee). He was illiterate and would have had no exposure to Mooney or other ethnographers.
I've heard the same or similar stories from native students from different tribes (including Cherokee) when I used to teach Native American Studies.
I've heard the story of Awi Usdi ("Little Deer") from native educators and elders in the context of "this is why we are careful about how much we hunt."
I've also heard from white historians and anthropologists that the story is a bunch of bullshit in the finest "noble savage" tradition and that everyone knows Indians hunted megafauna to extinction, so they must have been unaware of the consequences. Take that for what it is worth.
More information on James Mooney and his "histories" and myths:
Project Gutenberg's online copy of Myths of the Cherokee
Elliot (1998). "Ethnography, Reform, and the Problem of the Real: James Mooney's Ghost-Dance Religion" American Quarterly.
Again anecdotal, but I've had several Cherokee people (educators and elders) tell me that what James Mooney collected was the stories that children were told, since he wasn't considered an insider or reliable. Anything of sacred or personal worth was not given to him. This actually makes me think that the stories in "Myths" are somewhat reliable, at least in the sense that they were common stories used to explain the world.