We consider the asymptotic key rates achieved in the simplest quantum key distribution protocols, namely, the BB84 and the six-state protocols when nonuniform noise is present in the system. We first observe that higher qubit error rates do not necessarily imply lower key rates. Second, we consider protocols with advantage distillation and show that it can be advantageous to use the basis with higher quantum bit error rate for the key generation. We then discuss the relation between advantage distillation and entanglement distillation protocols. We show that applying advantage distillation to a string of bits formed by the outcomes of measurements in the basis with a higher quantum bit error rate is closely connected to the two-to-one entanglement distillation protocol of Deutsch-Ekert-Jozsa-Macchiavello-Popescu-Sanpera [Phys. Rev. Lett. 77, 2818 (1996)PRLTAO0031-900710.1103/PhysRevLett.77.2818]. Finally, we discuss the implications of these results for implementations of quantum key distribution.