Recent analyses suggest that the number of species in a clade often increases rapidly at first, but that diversification subsequently slows, apparently as species fill ecological space. Support for diversity dependence comes largely from the failure of species richness to increase with clade age in some analyses of contemporary diversity. However, clades chosen for analysis generally are named taxa and thus are not selected at random. To avoid this potential bias, we analyzed the numbers of species and estimated ages of 150 pairs of sister clades established by dispersal of ancestral species between the Oriental and African biogeographic regions. The observed positive exponential relationship between clade size and age suggests that species diversify within clades without apparent limit. If this were true, the pattern of accumulation of sister-clade pairs with increasing age would be consistent with the random decline and extinction of entire clades, maintaining an overall balance in species richness. This “pulse” model of diversification is consistent with the fossil record of most groups and reconciles conflicting evidence concerning diversity dependence of clade growth.