Because of the Unruh effect, accelerated and inertial observers differ in their description of a given quantum state. The implications of this effect are explored for the entropy assigned by such observers to localized objects that may cross the associated Rindler horizon. It is shown that the assigned entropies differ radically in the limit where the number of internal states n becomes large. In particular, the entropy assigned by the accelerated observer is a bounded function of n. General arguments are given along with explicit calculations for free fields. The implications for discussions of the generalized second law and proposed entropy bounds are also discussed.