We present high signal-to-noise ratio, precise Y JH photometry and Y band (0.957-1.120 mu m) spectroscopy of HD 1160 B, a young substellar companion discovered from the Gemini NICI Planet Finding Campaign using the Subaru Coronagraphic Extreme Adaptive Optics instrument and the Gemini Planet Imager. HD 1160 B has typical mid-M dwarf-like infrared colors and a spectral type of M5.5(-0.5)(+1.0), where the blue edge of our Y band spectrum rules out earlier spectral types. Atmospheric modeling suggests HD 1160 B has an effective temperature of 3000-3100 K, a surface gravity of log g - 4-4.5, a radius of. 1.55 +/- 0.10 R-J, and a luminosity of log L/L circle dot - 2.76 +/- 0.05. Neither the primary's Hertzspring-Russell diagram position nor atmospheric modeling of HD 1160 B show evidence for a subsolar metallicity. Interpretation of the HD 1160 B spectroscopy depends on which stellar system components are used to estimate the age. Considering HD 1160 A, B and C jointly, we derive an age of 80-125 Myr, implying that HD 1160 B straddles the hydrogen-burning limit (70-90 M-J) If we consider HD 1160 A alone, younger ages (20-125 Myr) and a brown dwarf-like mass (35-90 M-J) are possible. Interferometric measurements of the primary, a precise Gaia parallax, and moderate-resolution spectroscopy can better constrain the system's age and how HD 1160 B fits within the context of (sub) stellar evolution.