Bulk heterojunction solar cells (BHJs) based on poly[N-9″-hepta-decanyl-2,7-carbazole-alt-5,5-(4′,7′-di-2-thienyl-2′,1′,3′-benzothiadiazole)] (PCDTBT) can have internal quantum efficiencies approaching 100% but require active layers that are too thin to absorb more than ∼70% of the above band gap light. When the active layer thickness is increased so that the cell absorbs more light, the fill factor and open circuit voltage decrease rapidly, so that the overall power conversion efficiency decreases. We find that hole-traps in the polymer, which we characterize using space-charge limited current measurements, play an important role in the performance of PCDTBT-based BHJs and may limit the active layer thickness. Recombination due to carrier trapping is not often considered in BHJs because it is not believed to be a dominant loss mechanism in the “fruit-fly” P3HT system. Furthermore, we show that in contrast to P3HT, PCDTBT has only weak short-range molecular order, and that annealing at temperatures above the glass transition decreases the order in the π–π stacking. The decrease in structural order is matched by the movement of hole-traps deeper into the band gap, so that thermal annealing worsens hole transport in the polymer and reduces the efficiency of PCDTBT-based BHJs. These findings suggest that P3HT is not prototypical of the new class of high efficiency polymers, and that further improvement of BHJ efficiencies will necessitate the study of high efficiency polymers with low structural order.