Sloppy models are complex nonlinear models with outcomes that are significantly affected by only a small subset of parameter combinations. Despite forming an important universality class and arising frequently in practice, formal and systematic explanations of sloppiness are lacking. By unifying geometric interpretations of sloppiness with Chebyshev approximation theory, we offer such an explanation, and show how sloppiness can be described explicitly in terms of model smoothness. Our approach results in universal bounds on model predictions for classes of smooth models, and our bounds capture global geometric features that are intrinsic to their model manifolds. We illustrate these ideas using three disparate models: exponential decay, reaction rates from an enzyme-catalysed chemical reaction, and an epidemiology model of an infected population.