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 $kappa$ 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 datasets, 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 $sim 10$%. 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. $T_e$(O III), $T_e$(S III), and $T_e$(Ar III), cannot be conciliated by invoking $kappa$ 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 $kappaapprox 10$, which would then lead to significant abundance enhancements for these ions. An analytic formula is proposed to generate accurate $kappa$-averaged excitation rate coefficients (better than 10% for $kappa geq 5$) from temperature tabulations of the Maxwell-Boltzmann effective collision strengths.
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