Dielectric spectra were measured for 1,4-polybutadiene (PBD) at various temperatures and pressures corresponding to a constant value of the relaxation time, tau(alpha), for the local segmental dynamics (alpha-process). Given the relationship of the Johari-Goldstein secondary relaxation to the alpha-process, it is of interest to determine whether the frequency separation of the Johari-Goldstein secondary relaxation and the alpha relaxation remains essentially constant under isochronal conditions. We find for PBD this is not the case; the JG relaxation peak moves systematically to higher frequencies at constant tau(alpha) with increasing temperature and pressure. We show using molecular dynamics simulations that the behavior of PBD, which differs from that reported previously for molecular liquids, is a consequence of the torsional inflexibility of the polymer backbone. This accentuates the effect of constraints from local intermolecular barriers, with consequent deviation in the response of the JG and alpha relaxations due to their different dynamic length scales. Thus, although the Johari-Goldstein relaxation is related to and may even evolve into the structural relaxation associated with the glass transition, its response is not universal but rather depends to some extent on the chemical structure.