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Aim: The climate variability hypothesis proposes that species subjected to wide variation in climatic conditions will evolve wider niches, resulting in larger distributions. We test this hypothesis in tropical plants across a broad elevational gradient; specifically, we use a species‐level approach to evaluate whether elevational range sizes are explained by the levels of thermal variability experienced by species. Location: Central Andes. Time Period: Present day. Taxon: Woody plants. Methods: Combining data from 479 forest plots, we determined the elevational distributions of nearly 2300 species along an elevational gradient (~209–3800 m). For each species, we calculated the maximum annual variation in temperature experienced across its elevational distribution. We used phylogenetic generalized least square models to evaluate the effect of thermal variability on range size. Our models included additional covariates that might affect range size: body size, local abundance, mean temperature and total precipitation. We also considered interactions between thermal variability and mean temperature or precipitation. To account for geometric constraints, we repeated our analyses with a standardized measure of range size, calculated by comparing observed range sizes with values obtained from a null model. Results: Our results supported the main prediction of the climate variability hypothesis. Thermal variability had a strong positive effect on the range size, with species exposed to higher thermal variability having broader elevational distributions. Body size and local abundance also had positive, yet weak effects, on elevational range size. Furthermore, there was a strong positive interaction between thermal variability and mean annual temperature. Main Conclusions: Thermal variability had an overriding importance in driving elevational range sizes of woody plants in the Central Andes. Moreover, the relationship between thermal variability and range size might be even stronger in warmer regions, underlining the potential vulnerability of tropical montane floras to the effects of global warming. [ABSTRACT FROM AUTHOR]