The question whether the dynamics of hydrated proteins changes with temperature on crossing the glass transition temperature like that found in conventional glassformers is an interesting one. Recently, we have shown that a change of temperature dependence of the mean square displacement (MSD) at Tg is present in proteins solvated with bioprotectants, such as sugars or glycerol with or without the addition of water, coexisting with the dynamic transition at a higher temperature Td. The dynamical change at Tg is similar to that in conventional glassformers at sufficiently short times and low enough temperatures, where molecules are mutually caged by the intermolecular potential. This is a general and fundamental property of glassformers which is always observed at or near Tg independent of the energy resolution of the spectrometer, and is also the basis of the dynamical change of solvated proteins at Tg. When proteins are solvated with bioprotectants they show higher Tg and Td than the proteins hydrated by water alone, due to the stabilizing action of excipients, thus the observation of the change of T-dependence of the MSD at Tg is unobstructed by the methyl-group rotation contribution at lower temperatures [S. Capaccioli, K. L. Ngai, S. Ancherbak, and A. Paciaroni, J. Phys. Chem. B 116, 1745 (2012)]. On the other hand, in the case of proteins hydrated by water alone unambiguous evidence of the break at Tg is hard to find, because of their lower Tg and Td. Notwithstanding, in this paper, we provide evidence for the change at Tg of the T-dependence of proteins hydrated by pure water. This evidence turns out from (i) neutron scattering experimental investigations where the sample has been manipulated by either full or partial deuteration to suppress the methyl-group rotation contribution, and (ii) neutron scattering experimental investigations where the energy resolution is such that only motions with characteristic times shorter than 15 ps can be sensed, thus shifting the onset of both the methyl-group rotation and the dynamic transition contribution to higher temperatures. We propose that, in general, coexistence of the break of the elastic intensity or the MSD at Tg with the dynamic transition at Td in hydrated and solvated proteins. Recognition of this fact helps to remove inconsistency and conundrum encountered in interpreting data of hydrated proteins that thwart progress in understanding the origin of the dynamic transition.