The impact of radiation damage on silicon photomultiplier (SiPM) performance is essential in evaluating their suitability for use in long-term applications and/or high-radiation environments. In this work multiple SiPMs are tested from Hamamatsu, Ketek, and SensL. SiPMs are exposed separately to both gamma ray and fast neutron fields up to 1000 kRad and 1011 neutrons/cm2 respectively. Characteristics of interest are pulse shape discrimination (PSD) with organic scintillators, energy resolution, noise, and signal amplitude. It is found that PSD is more dramatically impacted by radiation damage than energy resolution, and significant increases in noise levels are observed at the irradiation levels employed. Attenuation of the signal amplitude is only significant for some SiPMs, notably those that produce the largest charge integrals. As expected, performance in larger SiPMs deteriorates faster because noise levels increase more drastically compared to smaller SiPMs. Neutron activation is also shown to not be the primary factor in performance degradation.