We identify [Se iii] 1.0994 μm in the planetary nebula (PN) NGC 5315 and [Kr vi] 1.2330 μm in three PNe from spectra obtained with the Folded-Port InfraRed Echellette (FIRE) spectrometer on the 6.5 m Baade Telescope. Se and Kr are the two most widely detected neutron-capture elements in astrophysical nebulae, and can be enriched by s-process nucleosynthesis in PN progenitor stars. The detection of [Se iii] 1.0994 μm is particularly valuable when paired with observations of [Se iv] 2.2864 μm, as it can be used to improve the accuracy of nebular Se abundance determinations, and allows Se ionization correction factor (ICF) schemes to be empirically tested for the first time. We present new effective collision strength calculations for Se2+ and Kr5+, which we use to compute ionic abundances. In NGC 5315, we find that the Se abundance computed from Se3+/H+ is lower than that determined with ICFs that incorporate Se2+/H+. We compute new Kr ICFs that take Kr5+/H+ into account, by fitting correlations found in grids of Cloudy models between Kr ionic fractions and those of more abundant elements, and use these to derive Kr abundances in four PNe. Observations of [Se iii] and [Kr vi] in a larger sample of PNe, with a range of excitation levels, are needed to rigorously test the ICF prescriptions for Se and our new Kr ICFs. This paper includes data obtained with the 6.5-m Magellan Telescopes at Las Campanas Observatory, Chile.