In superconducting-normal-metal sandwich structures in which the pair potential has abrupt spatial changes, Andreev scattering occurs in which electronlike and holelike excitations are intermixed. The Tomasch effect, Rowell-McMillan oscillations, and de Gennes-Saint James bound states result from interference of these excitations within a single film. In this paper we discuss compound resonances which can occur in both films of a sandwich structure in which each film is clean and has a thickness on the order of its coherence length. Two of the compound-resonance effects have the following interpretations: (1) Below the energy gap, the de Gennes-Saint James bound states split into energy bands. (2) Above the energy gap, resonances occurring with different frequencies result in a beating of the oscillations of the density of states as a function of energy. A number of previously puzzling or poorly understood features of published experimental data are shown to be consistent with compound-resonance effects. Suggestions are given for systematic experiments. © 1980 The American Physical Society.