Multinuclear spin-lattice relaxation rate measurements are reported over wide ranges of temperature, pressure, and concentration in undercooled aqueous solutions of tetraethylammonium bromide (Et(4)NBr). These dissolved organic cations with apolar surface groups provide model systems to investigate their influence upon the dynamics of the water molecules in the hydration shell as well as the influence of the hydration shell upon the dynamics of the dissolved hydrophobic ions. The isobaric relaxation time curves of all nuclei investigated show pronounced minima, indicating a strong slowing down of the dynamics of the solvent molecules in the hydration shell as well as of the dissolved hydrophobic ions. A VTF-like temperature dependence and corresponding scaling behavior allow glass transition temperatures to be deduced which signify the onset of motional arrest in these metastable solutions. Segmental motions of the aliphatic chains of the organic cations are discussed in terms of correlation times, and corresponding order parameters are given characterizing their spatial restrictions. At low temperatures all proton relaxation rates become equal to those of the methyl groups due to strong cross-relaxation effects, indicating that methyl groups act as relaxation sinks in these systems.