We present the results of a multiplicity survey for a magnitude-limited sample of 31 classical Be stars conducted with the Navy Precision Optical Interferometer and the Mark III Stellar Interferometer. The interferometric observations were used to detect companions in 10 previously known binary systems. For two of these sources (66 Oph and β Cep) new orbital solutions were obtained, while for a third source (υ Sgr) our observations provide the first direct, visual detection of the hot companion to the primary star. Combining our interferometric observations with an extensive literature search, we conclude that an additional four sources (o Cas, 15 Mon, β Lyr, and β Cep) also contain wider binary components that are physical companions to the narrow binaries, thus forming hierarchical multiple systems. Among the sources not previously confirmed as spectroscopic or visual binaries, BK Cam was resolved on a number of nights within a close physical proximity of another star with relative motion possibly suggesting a physical binary. Combining our interferometric observations with an extensive literature search, we provide a detailed listing of companions known around each star in the sample, and discuss the multiplicity frequency in the sample. We also discuss the prospects for future multiplicity studies of classical Be stars by long-baseline optical interferometry.