Natural sources of radiation are not strong enough to give anyone radiation poisoning. The circumstances that create enough radioactivity for that are highly artificial (nuclear reactors, nuclear weapons, concentrated isotopes, particle accelerators, X-ray machines, etc.).* You need a very high level of radiation, delivered over a relatively short amount of time, to suffer from the symptoms of radiation sickness.

One can contract cancer from exposure to natural radioactivity — notably by spending a lot of time in mines that contain significant amounts of uranium in their ores. Uranium ores exude radon and radon daughter products, and with inadequate ventilation these radioactive gases can concentrate at relatively high levels. Over time, this is a chronic lung cancer risk.

This was first noticed in the 16th century, when hard-rock miners in Joachimsthal (a uranium-rich region of what is now the Czech Republic) were noticed to have come down with a "Bergkrankheit" (mountain sickness) at higher rates than others. Only in the late-19th century (1879) was it understood to be a form of lung cancer. The exact mechanism of course required the discovery of radioactivity, and more careful study. (See Egon Lorentz, "Radioactivity and Lung Cancer: A Critical Review of Lung Cancer in the Miners of Schneeberg and Joachimsthal," 1944.)

^(*Whenever I write about this, someone feels the need to point out that over a billion years ago, it was possible to have natural nuclear reactors, like the one at Oklo. That's interesting and all, but irrelevant for this question, since the conditions for this only worked well before human beings had evolved. The Oklo "reactor" only functioned because the isotope ratio of U-235:U-238 was more favorable than it has been for hundreds of millions of years.)