Studies of symmetry-breaking are reported for self-consistent field wave functions applied to three homonuclear diatomics: F-2(+), O-2(+), and Cr-2. The results complement and extend existing reports on symmetry-breaking in three main ways. Two of these aspects concern manifestations of symmetry-breaking. First, calculations are reported that show energies at long interatomic separations which are lower than the sum of the atomic limits. This artifactual behavior, observed both with Hartree-Fock and Kohn-Sham density functional theory, appears to be largely associated with insufficiently flexible atomic orbital basis sets. Second, potential curves with spurious maxima and minima are found, associated with changes in spin-coupling. The third and final aspect, concerns the origin of symmetry-breaking. It is argued that the simplest way to understand symmetry breaking is as a consequence of the incorrect asymptotic behavior of limited wave functions at long interatomic distances.