The method of indirect demonstration is used to investigate if compaction disequilibrium can account for high overpressures that occur in Mesozoic and older basin formations. First the equations governing compaction disequilibrium are analysed for the factors controlling overpressure levels. Then limiting values of these control parameters are sought which favour high fluid pressures. The analysis shows why 'close-to-lithostatic fluid pressures' in pre-Cenozoic basin units are difficult to attain by compaction disequilibrium alone. Subsequently, the limiting favourable conditions are used in a series of generic numerical model experiments. The experiments serve as templates to construct the upper bounds of overpressures due to sediment loading for most geological settings including those where shale seals have developed. Two regional examples are studied in some detail. It is shown that observed overpressures in Mesozoic strata on the Scotian Shelf can be explained by compaction disequilibrium, but require the limiting values assigned to the properties of shale. For the Central North Sea Graben these limiting conditions are not sufficient, providing evidence for an active role of other pressure-generating mechanisms.