We show that upper bounds exist on the density of neutrons that can be moderated to a specified energy from an intense pulsed source of fast neutrons. Expressions are derived for the maximum density in the following cases: (a) a uniform infection of fast neutrons into an infinite moderator, (b) a localized central source in a finite heavy atom moderator, and (c) a point source in an infinite hydrogenous moderator. Correspondingly, upper bounds are given for the rates of single- and multiple-resonance neutron capture.