The M-squared (M2) parameter for defining laser beam quality is a convenient metric to characterize the variation of the beam size and far field divergence of a “real” propagated optical beam, compared with that of an ideal Gaussian beam of the same wavelength. However, it can be problematic to use this parameter solely to characterize an input beam for propagation simulation software. Similar to RMS wavefront error or Strehl ratio, which can be used to define image quality, but do not characterize the shape of the wavefront, different factors can result in beams with identical M2 values, but very different propagation behavior. Beams that differ due to aberrations, non-Gaussian amplitude envelopes, and/or partial spatial coherence may have similar or identical M2 values, but very different far-field and/or near-field intensity and/or phase distributions. The situation is complicated further if the beam encounters non-ideal optics. In this paper, we investigate a number of beams that all have (approximately) the same M2. While M2 is invariant for propagation through an ideal optical system, we show that when an optical system introduces aberrations, it can alter different beams with the same, non-unity M2 in ways that differ significantly from one beam to another.