The current-voltage (I-V) characteristics at 77 K are presented up to a pressure of 16 kbar for an AlAs/GaAs single-barrier structure and an AlAs/GaAs double-barrier structure behaving like a single-barrier structure. The current is first seen to rise with pressure and then fall at higher pressure close to and beyond the type I to type II transition-a signature for the involvement of the X profile. We present one model to account for this behaviour in both the low-pressure regime and the new high-pressure regime. For the latter we stress the importance of the pinning of the Fermi level in the contacts by the lowest confined X state in the AlAs region, which leads to band bending and the formation of depletion barriers in the contacts. Additionally, we contrast the behaviour of these indirect-gap barrier structures with that of a direct-gap Al0.35Ga0.64As/GaAs structure over the same pressure range, and discuss the possible origin of the falling background current in AlAs/GaAs double-barrier structures displaying X-X interwell and mixed X- Gamma resonances in the high-pressure regime.