The starburst galaxy NGC 253 was observed with the Energetic Gamma Ray Experiment Telescope (EGRET) aboard the Compton Gamma Ray Observatory (CGRO) satellite. We obtain a 2 sigma upper limit to the gamma-ray emission above 100 MeV of 8 x 10(-8) photons cm(-2) s(-1). Because of their large gas column densities and supernova rates, nearby starburst galaxies were predicted to have gamma-ray fluxes detectable by EGRET. Our nondetection of gamma-rays from NGC 253 motivates us to reexamine in detail the premise of supernova acceleration of cosmic rays and the effect of enhanced cloud densities, photon densities, and magnetic fields on the high-energy spectra of galaxies. By modeling the expected gamma-ray and synchrotron spectra from NGC 253, we find that up to 20% of the energy from supernovae is transferred to cosmic rays in the starburst, which is consistent with supernova acceleration models. Our calculations match the EGRET and radio data well with a supernova rate of 0.08 yr(-1), a magnetic field B greater than or similar to 5.5 x 10(-5) G, a density n similar to 300 cm(-3), a photon density U-ph similar to 200 eV cm(-3), and an escape timescale tau(0) less than or similar to 10 Myr.