The hypothesis of a characteristic rate of exponential radiation for the Insecta is tested for forty-nine of the most diverse family-level taxa in the Coleoptera, Diptera, and Hymenoptera. Data used are estimates of Recent species diversity and age of first appearance in the fossil record, as in the original statement of the hypothesis by Stanley (1979). There is no evidence of a characteristic radiation rate for the insects. A more exact method is proposed for identifying clades that are radiating exponentially. When applied to Stanley's data for the mammalian family Bovidae, it indicates that the Bovidae are not radiating exponentially at a characteristic rate. Future research might emphasize the reliable identification of clades that are radiating exponentially, prior to drawing conclusions about rates that might characterize higher taxa.--Modified journal abstract.