We present an analysis of elemental abundances of He, N, O, Ne, S, and Ar in Magellanic Cloud planetary nebulae (PNe) and focus initially on 14 PNe in the Small Magellanic Cloud (SMC). We derive the abundances from a combination of deep, high-dispersion optical spectra, as well as mid-infrared (IR) spectra from the Spitzer Space Telescope. A detailed comparison with prior SMC PN studies shows that significant variations in relative emission-line flux determinations among the authors, lead to systematic discrepancies in derived elemental abundances between studies that are gsim0.15 dex, in spite of similar analysis methods. We use ionic abundances derived from IR emission lines, including those from ionization stages not observable in the optical, to examine the accuracy of some commonly used recipes for ionization correction factors (ICFs). These ICFs, which were developed for ions observed in the optical and ultraviolet, relate ionic abundances to total elemental abundances. We find that most of these ICFs work very well even in the limit of substantially sub-solar metallicities, except for PNe with very high ionization. Our abundance analysis shows enhancements of He and N that are predicted from prior dredge-up processes of the progenitors on the asymptotic giant branch (AGB), as well as the well-known correlations among O, Ne, S, and Ar that are little affected by nucleosynthesis in this mass range. We identify MG 8 as an interesting limiting case of a PN central star with a ≈3.5 M sun progenitor in which hot-bottom burning did not occur in its prior AGB evolution. We find no evidence for O depletion in the progenitor AGB stars via the O-N cycle, which is consistent with predictions for lower-mass stars. We also find low S/O ratios relative to SMC H II regions, with a deficit comparable to what has been found for Galactic PNe. Finally, the elemental abundances of one object, SMP-SMC 11, are more typical of SMC H II regions, which raises some doubt about its classification as a PN.