The influence of sleep on motor skill consolidation has been a research topic of increasing interest. In this study, we distinguished general skill learning from sequence-specific learning in a probabilistic implicit sequence learning task (alternating serial reaction time) in young and old adults before and after a 12-h offline interval which did or did not contain sleep (p.m.-a.m. and a.m.-p.m. groups, respectively). The results showed that general skill learning, as assessed via overall reaction time, improved offline in both the young and older groups, with the young group improving more than the old. However, the improvement was not sleep-dependent, in that there was no difference between the a.m.-p.m. and p.m.-a.m. groups. We did not find sequence-specific offline improvement in either age group for the a.m.-either p.m. or p.m.-a.m. groups, suggesting that consolidation of this kind of implicit motor sequence learning may not be influenced by sleep.