A number of situations where it has been claimed that moderate radiation doses cause leukemia or other cancers are carefully reviewed. We look at cases in the United States, Great Britain, and the Soviet Union. Usually it can be demonstrated that there is an alternative, more probable, explanation for the effect seen. In several cases the authors of the papers have fallen into statistical traps. The most frequent is a posteriori selection of cohort boundaries in both space and time: a trap illustrated dramatically by Feynman. The next most common trap is to arbitrarily select one out of many ways of looking at the data, against which we were warned by Tippett. Several cohorts are compared with respect to the number of persons at risk, average dose, and the number of cancers expected. Of these, only the cohort of A-bomb survivors in Japan and the recently unclassified data on the very large occupational doses for early Soviet nuclear workers at Chelyabinsk provide evidence of clearly visible excess cancers.