The isostructural dimers of the 1,4-phenylene-bridged bis-1,2,3,5-dithia- and bis-1,2,3,5-diselenadiazolyl diradicals 1,4-S/Se are small band gap semiconductors. The response of their molecular and solid state electronic structures to pressure has been explored over the range 0-10 GPa. The crystal structures, which consist of cofacially aligned (pancake) p-dimers packed into herringbone arrays, experience a continuous, near-isotropic compression. While the intramolecular covalent EE (E = S/Se) bonds remain relatively unchanged with pressurization, the intradimer E...E separations are significantly shortened. Molecular and band electronic structure calculations using density functional theory methods indicate that compression of the pi-dimers leads to a widening of the gap Delta E between the highest occupied and lowest unoccupied molecular orbitals of the dimer, an effect that offsets the expected decrease in the valence-to-conduction band gap E-g occasioned by pressure-induced spreading of the valence and conduction bands. Consistent with the predicted consequences of this competition between intra- and interdimer interactions, variable temperature high pressure conductivity measurements reveal at best an order-of-magnitude increase in conductivity with pressure for the two compounds over the pressure range 0-10 GPa. While a small reduction in the thermal activation energy E-act with increasing pressure is observed, extrapolation of the rate of decrease suggests a projected onset of metallization (E-act approximate to 0) in excess of 20 GPa.