Theoretical N III electron density-sensitive emission-line ratios involving intercombination transitions, derived using recent calculations of electron impact excitation rates and oscillator strengths, are presented for R1 = I(1754.0 angstrom)/I(1749.7 angstrom), R2 = I(1752.2 angstrom)/I(1749.7 angstrom), R3 = I(1748.6 angstrom)/I(1749.7 angstrom), and R4 = I(1746.8 angstrom)/I(1749.7 angstrom). The observed values of R1, R2, and R3 for several gaseous nebulae, measured from high-resolution spectra obtained with the International Ultraviolet Explorer (IUE) satellite, imply electron densities that are compatible. However, values of N. derived from the R4 ratio are up to several orders of magnitude smaller than those deduced from R1, R2, and R3, which is probably due to the N III 1746.8 angstrom line being blended with Fe II 1746.8 angstrom. The electron densities deduced from the N III diagnostics are, in some objects, much larger than those estimated from line ratios in nebular ions such as O III, but are in good agreement with values deduced from the chromospheric C II intercombination transitions at approximately 2325 angstrom. These results suggest that in these nebulae the N III emission may also be chromospheric in origin.