Estimates of evolutionary diversification rates – speciation and extinction – have been used extensively to explain global biodiversity patterns. Many studies have analysed diversification rates derived from just two pieces of information: a clade's age and its extant species richness. This "age-richness rate" (ARR) estimator provides a convenient shortcut for comparative studies, but makes strong assumptions about the dynamics of species richness through time. We demonstrate that use of the ARR estimator in comparative studies is problematic on both theoretical and empirical grounds. We prove mathematically that ARR estimates are non-identifiable: there is no information in the data for a single clade that can distinguish a process with positive net diversification from one where net diversification is zero. Using palaeontological time series, we demonstrate that the ARR estimator has no predictive ability for real datasets. These pathologies arise because the ARR inference procedure yields "point estimates" that have been computed under a saturated statistical model with zero degrees of freedom. Although ARR estimates remain useful in some contexts, they should be avoided for comparative studies of diversification and species richness.