Studies of liquid water in its supercooled region have helped us better understand the structure and behavior of water. Bulk water freezes at its homogeneous nucleation temperature (approximately 235 K), but protein hydration water avoids this crystallization because each water molecule binds to a protein. Here, we study the dynamics of the hydrogen bond (HB) network of a percolating layer of water molecules and compare the measurements of a hydrated globular protein with the results of a coarse-grained model that successfully reproduces the properties of hydration water. Using dielectric spectroscopy, we measure the temperature dependence of the relaxation time of proton charge fluctuations. These fluctuations are associated with the dynamics of the HB network of water molecules adsorbed on the protein surface. Using Monte Carlo simulations and mean-field calculations, we study the dynamics and thermodynamics of the model. Both experimental and model analyses are consistent with the interesting possibility of two dynamic crossovers, (i) at approximately 252 K and (ii) at approximately 181 K. Because the experiments agree with the model, we can relate the two crossovers to the presence at ambient pressure of two specific heat maxima. The first is caused by fluctuations in the HB formation, and the second, at a lower temperature, is due to the cooperative reordering of the HB network.