The use of fullerenes with two or more adducts as acceptors has been recently shown to enhance the performance of bulk‐heterojunction solar cells using poly(3‐hexylthiophene) (P3HT) as the donor. The enhancement is caused by a substantial increase in the open‐circuit voltage due to a rise in the fullerene lowest unoccupied molecular orbital (LUMO) level when going from monoadducts to multiadducts. While the increase in the open‐circuit voltage is obtained with many different polymers, most polymers other than P3HT show a substantially reduced photocurrent when blended with fullerene multiadducts like bis‐PCBM (bis adduct of Phenyl‐C61‐butyric acid methyl ester) or the indene C60 bis‐adduct ICBA. Here we investigate the reasons for this decrease in photocurrent. We find that it can be attributed partly to a loss in charge generation efficiency that may be related to the LUMO‐LUMO and HOMO‐HOMO (highest occupied molecular orbital) offsets at the donor‐acceptor heterojunction, and partly to reduced charge carrier collection efficiencies. We show that the P3HT exhibits efficient collection due to high hole and electron mobilities with mono‐ and multiadduct fullerenes. In contrast the less crystalline polymer Poly[[9‐(1‐octylnonyl)‐9H‐carbazole‐2,7‐diyl]‐2,5‐thiophenediyl‐2,1,3‐benzothiadiazole‐4,7‐diyl‐2,5‐thiophenediyl (PCDTBT) shows inefficient charge carrier collection, assigned to low hole mobility in the polymer and low electron mobility when blended with multiadduct fullerenes.