Predictive frameworks of climate change extinction risk generally focus on the magnitude of climate change a species is expected to experience and the potential for that species to track suitable climate. A species' risk of extinction from climate change will depend, in part, on the magnitude of climate change the species experiences, its exposure. However, exposure is only one component of risk. A species' risk of extinction will also depend on its intrinsic ability to tolerate changing climate, its sensitivity. We examine exposure and sensitivity individually for two example taxa, terrestrial amphibians and mammals. We examine how these factors are related among species and across regions and how explicit consideration of each component of risk may affect predictions of climate change impacts. We find that species' sensitivities to climate change are not congruent with their exposures. Many highly sensitive species face low exposure to climate change and many highly exposed species are relatively insensitive. Separating sensitivity from exposure reveals patterns in the causes and drivers of species' extinction risk that may not be evident solely from predictions of climate change. Our findings emphasise the importance of explicitly including sensitivity and exposure to climate change in assessments of species' extinction risk.