We analyze the stellar populations present in M31 by using nine sets of adjacent Hubble Space Telescope NICMOS camera 1 and 2 fields with galactocentric distances ranging from 2' to 20'. These infrared observations provide some of the highest spatial resolution measurements of M31 to date; our data place tight constraints on the maximum luminosities of stars in the bulge of M31. The tip of the red giant branch is clearly visible at Mbol~-3.8, and the tip of the asymptotic giant branch (AGB) extends to Mbol~-5. This AGB peak luminosity is significantly fainter than previously claimed; through direct comparisons and simulations we show that previous measurements were affected by image blending. We do observe field-to-field variations in the luminosity functions, but simulations show that these differences can be produced by blending in the higher surface brightness fields. We conclude that the red giant branch of the bulge of M31 is not measurably different from that of the bulge of the Milky Way. We also find an unusually high number of bright bluish stars (7.3 arcmin-2), which appear to be Galactic foreground stars. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS 5-26555.