The Tropical Andes are an important global biodiversity hotspot, harbouring extraordinarily high richness and endemism. Although elevational richness and speciation have been studied independently in some Andean groups, the evolutionary and ecological processes that explain elevational richness patterns in the Andes have not been analysed together. Herein, we elucidate the processes underlying Andean richness patterns using glassfrogs (Centrolenidae) as a model system. Glassfrogs show the widespread mid-elevation diversity peak for both local and regional richness. Remarkably, these patterns are explained by greater time (montane museum) rather than faster speciation at mid-elevations (montane species pump), despite the recency of the major Andean uplift. We also show for the first time that rates of climatic-niche evolution and elevational change are related, supporting the hypothesis that climatic-niche conservatism decelerates species' shifts in elevational distributions and underlies the mid-elevation richness peak. These results may be relevant to other Andean clades and montane systems globally.