The dynamics of cis-1,4-polybutadiene (cis-1,4-PB) over a wide range of temperature and pressure conditions is explored by conducting atomistic molecular dynamics (MD) simulations with a united atom model on a 32-chain C128 cis-1,4-PB system. The local or segmental dynamics is analyzed in terms of the dipole moment time autocorrelation function (DACF) of the simulated polymer and its temperature and pressure variations, for temperatures as low as 195 K and pressures as high as 3 kbars. By Fourier transforming the DACF, the dielectric spectrum, epsilon* = epsilon' + i epsilon" = epsilon*omega, is computed and the normalized epsilon"/epsilon(max)" vs omega/omega(max) plot is analyzed on the basis of the time-temperature and time-pressure superposition principles. The relative contribution of thermal energy and volume to the segmental and chain relaxation processes are also calculated and evaluated in terms of the ratio of the activation energy at constant volume to the activation energy at constant pressure, Q(V)/Q(P). Additional results for the temperature and pressure dependences of the Rouse times describing terminal relaxation in the two polymers show that, in the regime of the temperature and pressure conditions covered here, segmental and chain relaxations are influenced similarly by the pressure and temperature variations. This is in contrast to what is measured experimentally [see, e.g., G. Floudas and T. Reisinger, J. Chem. Phys. 111, 5201 (1999); C. M. Roland et al.,J. Polym. Sci. Part B, 41, 3047 (2003)] for other, chemically more complex polymers that pressure has a stronger influence on the dynamics of segmental mode than on the dynamics of the longest normal mode, at least for the regime of temperature and pressure conditions covered in the present MD simulations.