ABSTRACT We present new near-infrared spectra, obtained at Gemini Observatory, for two Y dwarfs: WISE J035000.32−565830.2 (W0350) and WISEP J173835.52+273258.9 (W1738). A FLAMINGOS-2 R = 540 spectrum was obtained for W0350, covering , and a cross-dispersed Gemini near-infrared spectrograph R = 2800 spectrum was obtained for W1738, covering 0.993-1.087 m, 1.191-1.305 m, 1.589-1.631 m, and 1.985-2.175 m, in four orders. We also present revised YJH photometry for W1738, using new NIRI Y and J imaging, and a re-analysis of the previously published NIRI H-band images. We compare these data, together with previously published data for late-T and Y dwarfs, to cloud-free models of solar metallicity, calculated both in chemical equilibrium and with disequilibrium driven by vertical transport. We find that for the Y dwarfs, the non-equilibrium models reproduce the near-infrared data better than the equilibrium models. The remaining discrepancies suggest that fine-tuning the CH4/CO and NH3/N2 balance is needed. Improved trigonometric parallaxes would improve the analysis. Despite the uncertainties and discrepancies, the models reproduce the observed near-infrared spectra well. We find that for the Y0, W1738, , and log , and for the Y1, W0350, , and log . W1738 may be metal-rich. Based on evolutionary models, these temperatures and gravities correspond to a mass range for both Y dwarfs of 3-9 Jupiter masses, with W0350 being a cooler, slightly older, version of W1738; the age of W0350 is 0.3-3 Gyr, and the age of W1738 is 0.15-1 Gyr.