Electron impact excitation rates in Cl III, recently determined with the R-matrix code, are used to calculate electron temperature (T(e)) and density (N(e)) emission line ratios involving both the nebular (5517.7, 5537.9 A) and auroral (8433.9, 8480.9, 8500.0 A) transitions. A comparison of these results with observational data for a sample of planetary nebulae, obtained with the Hamilton Echelle Spectrograph on the 3-m Shane Telescope, reveals that the R(1) = I(5518 A)/I(5538 A) intensity ratio provides estimates of N(e) in excellent agreement with the values derived from other line ratios in the echelle spectra. This agreement indicates that R(1) is a reliable density diagnostic for planetary nebulae, and it also provides observational support for the accuracy of the atomic data adopted in the line ratio calculations. However the [Cl iii] 8433.9 A line is found to be frequently blended with a weak telluric emission feature, although in those instances when the [Cl iii] intensity may be reliably measured, it provides accurate determinations of T(e) when ratioed against the sum of the 5518 and 5538 A line fluxes. Similarly, the 8500.0 A line, previously believed to be free of contamination by the Earth's atmosphere, is also shown to be generally blended with a weak telluric emission feature. The [Cl iii] transition at 8480.9 A is found to be blended with the He i 8480.7 A line, except in planetary nebulae that show a relatively weak He i spectrum, where it also provides reliable estimates of T(e) when ratioed against the nebular lines. Finally, the diagnostic potential of the near-UV [Cl iii] lines at 3344 and 3354 A is briefly discussed.