Two peculiar features of the Saturnian satellite system, that is, the dense, N2-dominated atmosphere of Titan, and the very irregular shape of Hyperion, are probably related to each other. Numerical integrations show that most fragments ejected after the catastrophic breakup of a presumably larger and nearly-spherical proto-Hyperion fell onto Titan over a time scale of 103 yr, at impact speeds <4 km/s. Such impacts resulted into accretion of the constituent volatile materials into Titan's atmosphere. If proto- Hyperion was ≍1000 km in diameter, the entire current atmosphere may have been generated by this impact shower. Even for a smaller total mass of the impactors, any pre-existing atmosphere must have been chemically reprocessed, producing large amounts of N2 by shock effects. Impacts by proto-Hyperion fragments are also likely to have formed giant impact basins on Titan's surface.