In this paper, we investigate the nature of the coding mechanisms required to ensure strong secrecy over wiretap channels. Specifically, we analyze the limitations of capacity-based wiretap codes, i.e. wiretap codes that associate to each confidential message a subcode whose rate approaches the eavesdropper's channel capacity. For a wiretap channel with a noiseless main channel and a binary symmetric eavesdropper's channel, we show that secrecy-capacity achieving sequences of capacity-based wiretap codes cannot achieve the strong secrecy capacity. We also show that sequences of random capacity-based wiretap codes achieve strong secrecy rates provided the eavesdropper's channel is degraded with respect to the channel for which the codes were designed.