Using ab-initio photoionization models of giant H ii regions, we test methods for abundance determinations based on a direct measurement of the electron temperature, now possible even for moderate to high-metallicity objects. We find that, for metallicities larger than solar, the computed abundances deviate systematically from the real ones, generally by larger amounts for more metal-rich H ii regions. We discuss the reasons for this, and present diagrams allowing the reader to better understand the various factors coming into play. We briefly discuss less classical methods to derive abundances in metal-rich H ii regions. In particular, we comment on the interest of the oxygen and carbon recombination lines. We also show that, contrary to the case of giant H ii regions, the physical conditions in bright extragalactic planetary nebulae are such that their chemical composition can be accurately derived even at high metallicities. Thus, extragalactic planetary nebulae are promising potential probes of the metallicity of the interstellar medium in the internal parts of spiral galaxies as well as in metal-rich elliptical galaxies.