The tropics are the source of most biodiversity yet inadequate sampling obscures answers to fundamental questions about how this diversity evolves. We leveraged samples assembled over decades of fieldwork to study diversification of the largest tropical bird radiation, the suboscine passerines. Our phylogeny, estimated using data from 2389 genomic regions in 1940 individuals of 1287 species, reveals that peak suboscine species diversity in the Neotropics is not associated with high recent speciation rates but rather with the gradual accumulation of species over time. Paradoxically, the highest speciation rates are in lineages from regions with low species diversity, which are generally cold, dry, unstable environments. Our results reveal a model in which species are forming faster in environmental extremes but have accumulated in moderate environments to form tropical biodiversity hotspots.