We have identified two new near-infrared (NIR) emission lines in the spectra of planetary nebulae arising from heavy elements produced by neutron-capture reactions: [Te III] 2.1019 μm and [Br V] 1.6429 μm. [Te III] was detected in both NGC 7027 and IC 418, while [Br V] was seen in NGC 7027. The observations were obtained with the medium-resolution spectrograph Espectrógrafo Multiobjeto Infra-Rojo (EMIR) on the 10.4 m Gran Telescopio Canarias at La Palma, and with the high-resolution Immersion GRating INfrared Spectrometer (IGRINS) on the 2.7 m Harlan J. Smith telescope at McDonald Observatory. New calculations of atomic data for these ions, specifically A-values and collision strengths, are presented and used to derive ionic abundances of Te2+ and Br4+. We also derive ionic abundances of other neutron-capture elements detected in the NIR spectra, and estimate total elemental abundances of Se, Br, Kr, Rb, and Te after correcting for unobserved ions. Comparison of our derived enrichments to theoretical predictions from asymptotic giant branch (AGB) evolutionary models shows reasonable agreement for solar metallicity progenitor stars of ∼2–4 M ⊙. The spectrally isolated [Br V] 1.6429 μm line has advantages for determining nebular Br abundances over optical [Br III] emission lines that can be blended with other features. Finally, measurements of Te are of special interest because this element lies beyond the first peak of the s-process, and thus provides new leverage on the abundance pattern of trans-iron species produced by AGB stars.