  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 $\kappa\approx
10$, 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.
