We determine elemental abundances of He, N, O, Ne, S, and Ar in Magellanic Cloud planetary nebulae (PNe) using direct methods and a large set of observed ions, minimizing the need for ionization correction factors. In contrast to prior studies, we find a clear separation between Type I and non-Type I PNe in these low-metallicity environments, and no evidence that the O-N nucleosynthesis cycle is active in low-mass progenitors. We find that the S/O abundance ratio is anomalously low compared to H ii regions, confirming the ``sulfur anomaly'' found for Galactic PNe. We also found that Ne/O is elevated in some cases, raising the possibility that Ne yields in low-mass AGB stars may be enhanced at low metallicity.