The classic optical nebular diagnostics [N ii], [O ii], [O iii], [S ii], [S iii], and [Ar iii] are employed to search for evidence of non-Maxwellian electron distributions, namely κ distributions, in a sample of well-observed Galactic H ii regions. By computing new effective collision strengths for all these systems and A-values when necessary (e.g., S ii), and by comparing with previous collisional and radiative data sets, we have been able to obtain realistic estimates of the electron-temperature dispersion caused by the atomic data, which in most cases are not larger than ∼10%. If the uncertainties due to both observation and atomic data are then taken into account, it is plausible to determine for some nebulae a representative average temperature while in others there are at least two plasma excitation regions. For the latter, it is found that the diagnostic temperature differences in the high-excitation region, e.g., Te(O iii), Te(S iii), and Te(Ar iii), cannot be conciliated by invoking κ distributions. For the low-excitation region, it is possible in some, but not all, cases to arrive at a common, lower temperature for [N ii], [O ii], and [S ii] with κ ≈ 10, which would then lead to significant abundance enhancements for these ions. An analytic formula is proposed to generate accurate κ-averaged excitation rate coefficients (better than 10% for κ ⩾ 5) from temperature tabulations of the Maxwell–Boltzmann effective collision strengths.