Non-fullerene acceptors (NFAs) have enabled power conversion efficiencies exceeding 19% in organic solar cells (OSCs). However, the open-circuit voltage of OSCs remains low relative to their optical gap due to excessive non-radiative recombination, and this now limits performance. Here, an important aspect of OSC design is considered, namely management of the triplet exciton population formed after non-geminate charge recombination. By comparing the blends PM6:Y11 and PM6:Y6, it is shown that the greater crystallinity of the NFA domains in PM6:Y11 leads to a higher rate of triplet-triplet annihilation (TTA). This is attributed to the four times larger ground state dipole moment of Y11 versus Y6, which improves the long range NFA out-of-plane ordering. Since TTA converts a fraction of the non-emissive triplet states into bright singlet states, it has the potential to reduce non-radiative voltage losses. Through a kinetic analysis of the recombination processes under 1-Sun illumination, a framework is provided for determining the conditions under which TTA may improve OSC performance. If these could be satisfied, TTA has the potential to reduce non-radiative voltage losses by up to several tens of millivolts.