It has been proposed that sleep readies the brain for novel learning, and previous work has shown that sleep loss impairs the ability to encode new memories. In the present study, we examined if a daytime nap would increase mnemonic discrimination (MD) performance. MD is the ability to differentiate between memories that are similar but not identical. Participants performed the Mnemonic Similarity Task (MST) twice, once in the morning and once in the afternoon. The goal of this task is to distinguish stimuli that have been seen before from novel stimuli that are similar but not identical. After the morning MST, participants were randomly allocated into either a sleep or a wake group. The sleep group had a 2-hr nap opportunity, whereas the wake group spent a similar amount of time passively resting. All participants then performed a second MST in the afternoon with a novel set of images. Results did not show any support for increased MD ability after a nap. There was, however, a correlation showing that an increase in sleepiness between sessions predicted a decrease in MD performance. Future work must systematically examine how strong sleep manipulations that are needed for sleep to have an effect on encoding ability, as well as which kind of memory tasks that are sensitive to sleep manipulations. More knowledge about the relationship between sleep and the ability to differentiate similar memories from each other is important because impaired MD ability has previously been reported in various groups in which sleep disturbances are also common.