The basis for the conjecture that water's T-g may be 165+/-5 K [Velikov, Borick, and Angell, Science 294, 2335 (2001)] has been examined. It is shown that (i) differential scanning calorimetry (DSC) scans provided by Hallbrucker and Mayer [J. Phys. Chem. 91, 503 (1987)], and used as a basis for the conjecture, do not represent the heat capacity of the assumed, slow-cooled glassy water or of hyperquenched glassy water, and (ii) there is no fundamental requirement that the excess heat capacity show a peak at TT-g. On heating, the enthalpy of glasses produced by hyperquenching or rapid cooling begins to decrease at a much lower T than that of the glasses obtained by slow cooling. Annealing increases this temperature toward T-g, and the enthalpy decrease continues at T above T-g. In the enthalpy relaxation region, the diffusion coefficient of the hyperquenched glassy state is higher than that of a slow-cooled glassy state at a given T, and a local minimum in the DSC scan does not appear at T