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 1.0 < l mm < 1.7, and a cross-dispersed Gemini near-infrared spectrograph R = 2800 spectrum was obtained for W1738, covering 0.993–1.087 μm, 1.191–1.305 μm, 1.589–1.631 μm, and 1.985–2.175 μm, 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 CH$_4$/CO and NH$_3$/N$_2$ 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, T$_{eff}$ = 425 $\pm$ 25 K, and log $g$ = 4.0 $\pm$ 0.25, and for the Y1, W0350, T$_{eff}$ = 350 $\pm$ 25 K, and log $g$ = 4.0 $\pm$ 0.25. 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.