The evolution of a foam is determined by drainage flow of the continuous (liquid) phase and coarsening (aging) of the dispersed phase (gas bubbles). Free-drainage experiments with slow- and fast-coarsening gases show markedly different dynamics and elucidate the importance of the coupling of the two effects. Strong coarsening leads to drainage times that are shorter (accelerated drainage) and independent of the initial liquid content (self-limiting drainage). A model incorporating the physics of both drainage and diffusive coarsening shows quantitative agreement with experiment.