Abstract: We set an upper limit on the mass of a supersymmetric neutralino dark matter particle using the MicrOMEGAS and DarkSUSY software packages and the most recent constraints on relic density from combined Wilkinson Microwave Anisotropy Probe and Sloan Digital Sky Survey data. We explore several different possible scenarios within the minimal supersymmetric standard model, including coannihilation with charginos and sfermions and annihilation through a massive Higgs resonance, using low-energy mass inputs. We find that no coannihilation scenario is consistent with dark matter in observed abundance with a mass greater than 2.5 TeV for a $W$-ino-type particle or 1.8 TeV for a Higgsino-type. Contrived scenarios involving Higgs resonances with finely tuned mass parameters can allow masses as high as 34 TeV. The resulting gamma-ray energy distribution is not in agreement with the recent multi-TeV gamma-ray spectrum observed by H. E. S. S. originating from the center of the Milky Way. Our results are relevant only for dark matter densities resulting from a thermal origin.