The competition between impact erosion and impact supply of volatiles to planetary atmospheres can determine whether a planet or satellite accumulates an atmosphere. In the absence of other processes (e.g., outgassing), we find either that a planetary atmosphere should be thick, or that there should be no atmosphere at all. The boundary between the two extreme cases is set by the mass and velocity distributions and intrinsic volatile content of the impactors. We apply our model specifically to Titan, Callisto, and Ganymede. The impacting population is identified with comets, either in the form of stray Uranus-Neptune planetesimals or as dislodged Kuiper belt comets. Systematically lower impact velocities on Titan allow it to retain a thick atmosphere, while Callisto and Ganymede get nothing. Titan's atmosphere may therefore be an expression of a late-accreting, volatile-rich veneer. An impact origin for Titan's atmosphere naturally accounts for the high D/H ratio it shares with Earth, the carbonaceous meteorites, and Halley. It also accounts for the general similarity of Titan's atmosphere to those of Triton and Pluto, which is otherwise puzzling in view of the radically different histories and bulk compositions of these objects.