Data on lung cancer mortality rates vs. average radon concentration in homes for 1,601 U.S. counties are used to test the linear-no threshold theory. The widely recognized problems with ecological studies, as applied to this work, are addressed extensively. With or without corrections for variations in smoking prevalence, there is a strong tendency for lung cancer rates to decrease with increasing radon exposure, in sharp contrast to the increase expected from the theory. The discrepancy in slope is about 20 standard deviations. It is shown that uncertainties in lung cancer rates, radon exposures, and smoking prevalence are not important and that confounding by 54 socioeconomic factors, by geography, and by altitude and climate can explain only a small fraction of the discrepancy. Effects of known radon-smoking prevalence correlations--rural people have higher radon levels and smoke less than urban people, and smokers are exposed to less radon than non-smokers--are calculated and found to be trivial. In spite of extensive efforts, no potential explanation for the discrepancy other than failure of the linear-no threshold theory for carcinogenesis from inhaled radon decay products could be found.