Evolution provides many cases of apparent shifts in diversification associated with particular anatomical traits. Three general models connect these patterns to anatomical evolution: (i) elevated net extinction of taxa bearing particular traits, (ii) elevated net speciation of taxa bearing particular traits, and (iii) elevated evolvability expanding the range of anatomies available to some species. Trait-based diversification shifts predict elevated hierarchical stratigraphic compatibility (i.e., primitive→derived→highly derived sequences) among pairs of anatomical characters. The three specific models further predict (i) early loss of diversity for taxa retaining primitive conditions (elevated net extinction), (ii) increased diversification among later members of a clade (elevated net speciation), and (iii) increased disparity among later members in a clade (elevated evolvability). Analyses of 319 anatomical and stratigraphic datasets for fossil species and genera show that hierarchical stratigraphic compatibility exceeds the expectations of trait-independent diversification in the vast majority of cases, which was expected if trait-dependent diversification shifts are common. Excess hierarchical stratigraphic compatibility correlates with early loss of diversity for groups retaining primitive conditions rather than delayed bursts of diversity or disparity across entire clades. Cambrian clades (predominantly trilobites) alone fit null expectations well. However, it is not clear whether evolution was unusual among Cambrian taxa or only early trilobites. At least among post-Cambrian taxa, these results implicate models, such as competition and extinction selectivity/resistance, as major drivers of trait-based diversification shifts at the species and genus levels while contradicting the predictions of elevated net speciation and elevated evolvability models.