Facile control of the radiative and nonradiative properties of plasmonic nanostructures is of practical importance to a wide range of applications in the biological, chemical, optical, information, and energy sciences. For example, the ability to easily tune not only the plasmon spectrum but also the degree of coupling to light and/or heat, quality factor, and optical mode volume would aid the performance and function of nanophotonic devices and molecular sensors that rely upon plasmonic elements to confine and manipulate light at nanoscopic dimension. While many routes exist to tune these properties, identifying new approaches---especially when they are simple to apply experimentally---is an important task. Here, we demonstrate the significant and underappreciated effects that substrate thickness and dielectric composition can have upon plasmon hybridization as well as downstream properties that depend upon this hybridization. We find that even substrates as thin as ~10 nm can nontrivially mix free-space plasmon modes, imparting bright character to those that are dark (and vice versa) and, thereby, modifying the plasmonic density of states as well as the system's near- and far-field optical properties. A combination of electron energy-loss spectroscopy (EELS) experiment, numerical simulation, and analytical modeling is used to elucidate this behavior in the finite substrate-induced mixing of dipole, quadrupole, and octupole corner-localized plasmon resonances of individual silver nanocubes.