The authors report resonant tunnelling and thermionic emission measurements at pressures up to 9 kbar and temperatures down to 4.2 K, of p-i-n and n-i-n diodes containing a graded gap superlattice and double barrier structures, respectively, in which the wells are of GaAs and the barriers are of AlAs. The graded gap superlattice was grown by MBE and consists of five nominally 100 AA periods of GaAs/AlAs in which the 'mark-space' ratio of GaAs to AlAs is increased uniformly from 94 AA of GaAs, 6 AA of AlAs to 70 AA GaAs, 30 AA AlAs. The double barrier structures were grown by MOCVD with either approximately 70 AA wells and approximately 40 AA barriers or approximately 15 AA wells and approximately 50 AA barriers. The size of the NDR in the p-i-n structures increases with decreasing temperature as previously reported, the best samples exhibiting a ratio of maximum to minimum tunnelling current in excess of 2 at 4.2 K and 1 bar. On applying pressure, the size of the NDR at 4.2 K decreases slowly up to 5 kbar and more rapidly at higher pressures, almost vanishing by 8 kbar. A similar effect occurs in both directions of bias for the 43-72-40 AA double barrier structures, even though these have a ratio of maximum to minimum tunnelling current in excess of 10 at 4.2 K. In contrast, the 45-15-50 AA double barrier structures do not exhibit negative differential resistance, but thermionic emission measurements yield values for the barrier height, and its pressure dependence, of approximately 0.15 eV and -0.011 eV kbar-1 respectively. They take all these observations as evidence that X-like barrier states are strongly involved in the resonant tunnelling process, and that in general the pressure threshold for loss of NDR in samples grown in different ways, and with different ratios of peak to valley currents, is essentially a function only of well and barrier dimensions.