We investigate the charge-generation processes limiting the performance of low-offset organic bulk-heterojunction solar cells by studying a series of newly synthesized PBDB-T-derivative donor polymers whose ionisation energy (IE) is tuned via functional group (difluorination or cyanation) and backbone (thiophene or selenophene bridge) modifications. When blended with the acceptor Y6, the series present heterojunction donor-acceptor IE offsets (ΔEIE) ranging from 0.22 to 0.59 eV. As expected, small ΔEIE decrease nonradiative voltage losses but severely suppresses photocurrent generation. We explore the origin of this reduced charge-generation efficiency at low ΔEIE through a combination of opto-electronic and spectroscopic measurements and molecular and device-level modeling. We find that, in addition to the expected decrease in local exciton dissociation efficiency, reducing ΔEIE also strongly reduces the charge transfer (CT) state dissociation efficiency, demonstrating that poor CT-state dissociation can limit the performance of low-offset heterojunction solar cells.