In this work the translational dynamics of hydration water in protein matrices is investigated. It was shown that the determination of diffusion coefficents of water in protein matrices with field gradient NMR is non-trivially influenced by the effect of cross relaxation, the exchange of magnetization between water and protein protons. The model of Peschier, which describes how cross relaxation and diffusion affect the temporal evolution of the proton magnetization, was numerically solved. In order to elucidate the influence of cross relaxtion on diffusion measurements, typical NMR field gradient experiments have been simulated. A protocol has then been developed in order to analyse and elicit diffusion coefficents in a reliable way out of data from stimulated echo (STE) NMR experiments in high ( g ≥ 180 T/m) static field gradients (SFG). The findings show that either the effect of cross relaxation has been neclected in previous diffusion studies of hydration water or that the diffusion measurements have been conducted at the limit of the working regime of the used method, the pulsed field gradient (PFG) NMR. One goal of the present study was to test for the existence of the putative fragile-to-strong transition (FST) of supercooled hydration water at Tx = 223 K, and to establish the SFG-STE method for the study of translational dynamics of water in protein matrices. The present work devises explicitly the limit of the determination of diffusion coefficients of water in protein matrices. Evidence is given that it is not possible, using the samples and methods studied in this work, to study the translational dynamics in the vanity of the temperature where a change of temperature dependence of D is found ( Tx = 223 K).