Small molecule organic solar cells are becoming increasingly efficient through improved molecular design. However, there is still much to be understood regarding device operation. Here we study bilayer solar cells employing a 2,4-bis[4-(N,N-diisobutylamino)-2,6-dihydroxyphenyl] squaraine (SQ) donor and fullerene acceptor to probe the effect of donor layer thickness and a MoO₃ electron transport layer on device performance. The thickness of SQ is seen to drastically affect the open-circuit voltage (V_OC) and fill factor (FF), while the short circuit current is not altered significantly. The fact that the V_OC of the bilayers with thin (6 nm) donor layers shows a strong dependence on the material and workfunction of the anode cannot be explained with a model for a perfect bilayer. Recombination of electrons from C₆₀ at the anode contact has to be possible to understand the strong effect of the anode workfunction. Using numerical simulations and a simple two-diode model we show that the most likely interpretation of the observed effects is that for thin SQ layers, the roughness of the interface is high enough to allow electrons in the C₆₀ to tunnel through the SQ to recombine directly at the anode. Thicker SQ layers will block most of these recombination pathways, which explains the drastic dependence of V_OC on thickness. Bulk-heterojunction devices were also fabricated to illustrate the effect of anode material on the V_OC.