(Abridged) We examine standard methods of measuring nebular chemical abundances, including estimates based on direct T_e measurements, and also bright-line diagnostics. We use observations of 4 LMC HII regions whose ionizing stars have classifications ranging from O7 to WN3. We assume a 2-zone T_e structure to compute ionic abundances. We compare with photoionization models tailored to the properties of the individual objects, and emphasize the importance of correctly relating T_e in the two zones, which can otherwise cause errors of ~0.2 dex in abundance estimates. There are no spatial variations to within 0.1 - 0.15 dex in any of the objects, even one hosting 3 WR stars. Our data agree with the modeled R23 and S23 diagnostics of O and S. We present the first theoretical tracks for S23, which are in excellent agreement with a larger dataset. However, contrary to earlier suggestions, S23 is much more sensitive to the ionization parameter than is R23, because S23 does not sample S IV. We therefore introduce S234 = ([SII]+[SIII]+[SIV])/H-beta. Predicted and observed spatial variations in S234 are dramatically reduced in contrast to S23. The intensity of [SIV]10.5 microns is easily estimated from a simple relation between [SIV]/[SIII] and [OIII]/[OII]. This method of estimating S234 yields excellent agreement with our models, hence we give a theoretical calibration for S234. The double-valued structure of S23 and S234 remains an important problem as for R23, and presently we consider the S diagnostics reliable only at Z < 0.5 Z_sol. However, the slightly larger dynamic range and excellent compatibility with theoretical predictions suggest the S diagnostics to be more effective abundance indicators than R23.

Accepted to ApJ. 24 pages, 11 figures, uses emulateapj.sty