Non-thermal velocities obtained from the measurement of coronal Extreme Ultraviolet (EUV) linewidths have been consistently observed in solar EUV spectral observations and have been theorized to result from many plausible scenarios including wave motions, turbulence, or magnetic reconnection. Constraining these velocities can provide a physical limit for the available energy resulting from unresolved motions in the corona. We statistically determine a series of non-thermal velocity distributions from linewidth measurements of 390 emission lines from a wide array of elements and ionization states observed during the Solar Extreme Ultraviolet Research Telescope and Spectrograph 1991-1997 flights covering the spectral range 174-418 Å and a temperature range from 80,000 K to 12.6 MK. This sample includes 248 lines from active regions, 101 lines from quiet-Sun regions, and 41 lines were observed from plasma off the solar limb. We find a strongly peaked distribution corresponding to a non-thermal velocity of 19-22 km s-1 in all three of the quiet-Sun, active region, and off-limb distributions. For the possibility of Alfvén wave resonance heating, we find that velocities in the core of these distributions do not provide sufficient energy, given typical densities and magnetic field strengths for the coronal plasma, to overcome the estimated coronal energy losses required to maintain the corona at the typical temperatures working as the sole mechanism. We find that at perfect efficiency 50%-60% of the needed energy flux can be produced from the non-thermal velocities measured.