The structural phase behavior of high-quality single crystals of methylammonium lead iodide (CH3NH3PbI3 or MAPbI3) was revisited by combining Raman scattering and photoluminescence (PL) measurements under high hydrostatic pressure up to ca. 10 GPa. The single crystals were specially grown with the final thickness needed for pressure experiments, retaining their high quality due to a less invasive preparation procedure, which avoids sample thinning. Both PL and Raman spectra show simultaneous changes in their profiles that indicate the occurrence of three phase transitions subsequently at around 0.4, 2.7, and 3.3 GPa. At the second phase transition, the Raman spectra exhibit a pronounced reduction in the line width of the phonon modes of the inorganic cage, similar to the changes observed at the tetragonal-to-orthorhombic phase transition occurring at around 160 K but ambient pressure. This behavior is interpreted as evidence for the locking of the organic cations in the cage voids above 2.7 GPa due to the reduced volume and symmetry of the unit cell. At the third phase transition, reported here for the first time, the PL is greatly affected, whereas the Raman spectrum experiences only subtle changes related to a splitting of some of the peaks. This behavior may indicate a change mostly in the electronic structure with little effect on the crystal structure. Strikingly, the sharp Raman features observed at high pressures do not support amorphization of MAPbI3 with onset at 3 GPa, as claimed by most of the high-pressure (X-ray) literature. We interpret this apparent discrepancy in terms of the degree of disorder introduced at different length scales in the perovskite lattice by the pressure-induced freeze-out of the methylammonium cation motion.