From Wikipedia:
Although some areas continued an essentially Middle Mississippian culture until the first significant contact with Europeans, most areas had dispersed or were experiencing severe social stress by 1500. Along with the contemporary Anasazi, these cultural collapses coincide with the global climate change of the Little Ice Age.
So, it's a chicken or egg guess. Which came first? Perhaps cocoliztli was spread by rodents infesting these people BECAUSE of global cooling, when the natural food supply the rats relied on whithered up? This same disease (or the rodent carriers) spread Southward, reaching Mexico by 1545.
Yup. There's no reason why the Americas couldn't have had their own sort of 'Black Deaths.'