Earlier experimental investigations performed on a family of block copolymers formed from styrene and a homologous series of n-alkyl methacrylates revealed a strong dependence of thermodynamic compatibility between the two blocks on the length of the alkyl side chain of the methacrylate. Here we report the effect of hydrostatic pressure on the phase behavior of the same series of block copolymers as determined by in situ small-angle neutron scattering. We find that hydrostatic pressure is a very effective means of driving styrene/n-alkyl methacrylate block copolymers with intermediate side chains from the highly viscous ordered state to the fluid disordered state of the copolymer. Hence, for n ranging from 2 to 6 (ethyl to hexyl methacrylate), pressure induces mixing with an absolute value of the pressure coefficient of the order/disorder transition, dT(ODT)/dP, of up to 1.5degrees C/MPa (150degrees C/kbar). Similar results are obtained when the methacrylate block consists of a random sequence of short and long alkyl side chains with carefully chosen and predictable composition. In contrast, pressure suppresses mixing when the methacrylate block is composed of either very short (n = 1) or very long (n > 8) side chains. In terms of rheological properties, these results indicate that pressure applied at a constant temperature can be used to induce flow in some copolymers of this series. The ability to design such "baroplastic" behavior into commercially relevant thermoplastic elastomers would be highly advantageous from a processing standpoint.