Change in body mass with time has been considered for many clades, often with reference to Cope's rule, which predicts a tendency to increase in body size. A more general rule, namely increase in the range of body mass with time, is analyzed here for vertebrates. The log range of log vertebrate body mass is shown to increase linearly and highly significantly with the log of duration of clade existence. The resulting regression equations are used to predict the origin age, initial body mass, and subsequent dynamics of body mass range for primate clades such as the New World monkeys (Platyrrhini, 32 million years ago, initial mass of 1.7 kg) and the Anthropoidea (57 million years ago, initial mass of 0.12 kg), tested against the primate fossil record. Using these methods, other major primate clades such as Lemuriformes and Adapoidea are also estimated to have originated in the Tertiary (63 and 64 million years ago, respectively), with only the Plesiadapiformes originating in the Cretaceous (83 million years ago). Similarities of body mass range between primate and other vertebrate sister groups are discussed. Linear relationships of log range and log duration are considered with respect to Brownian processes, with the expected regression coefficients from the latter explored through simulations. The observed data produce regression coefficients that overlap with or are higher than those under Brownian processes. Overall, the analyses suggest the dynamics of vertebrate body mass range in morphologically disparate clades are highly predictable over many tens of million years and that the dynamics of phenotypic characteristics can assist molecular clock and fossil models in dating evolutionary events.