Studying the plasmonic properties of metallic nanoparticles at the individual nanostructure level is critical to our understanding of nanoscale metallic systems. Here we show how the presence of a nearby dielectric substrate modifies the energies of the plasmon modes of a metallic nanoparticle. The adjacent dielectric lifts the degeneracy of the dipole plasmon modes oriented parallel and perpendicular to the substrate, introducing a significant energy splitting that depends strongly on the permittivity of the substrate. This energy splitting can easily be misinterpreted as an anomalously broadened plasmon line shape for excitation of an individual nanoparticle with unpolarized light.