It is shown for two simple fixed-source theories with one stable excited state of the source that (1) a knowledge of the physical $S$-matrix elements of the theory is sufficient to decide if the state is associated with an elementary particle or a composite particle, and (2) the theory in which the particle is composite corresponds to the elementary-particle theory in the limit of vanishing wave function renormalization constant of the elementary particle. The distinction between elementary and composite particle is related to a generalization of Levinson's theorem, and it is made plausible that this generalization is valid in local field theory. In an Appendix, the result (2) is generalized, and an application to the dispersion-theoretic treatment of rearrangement collisions is suggested.