The mesoglobule formation and growth in an aqueous solution of poly(N-isopropylacrylamide) are investigated using time-resolved small-angle neutron scattering. Rapid pressure jumps initiate phase separation at both low (below 20 MPa) and high pressures (above 101 MPa). Measurements were carried out in a time range from 50 ms to similar to 1650 s after the jump, covering a large range of momentum transfers. The dehydration of the polymer at the coexistence line is much stronger at low pressures than at high pressures, which significantly affects the mechanism of phase separation: At low pressures, the diffusion-limited coalescence of the mesoglobules is strongly slowed down by the viscoelastic effect due to their dense shell. Moreover, the target pressure has a strong influence on the relative importance of these kinetic effects. In the high-pressure regime, the viscoelastic effect does not play a role, and diffusion-limited coalescence proceeds without hindrance.