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Whether adaptation is limited by the beneficial mutation supply is a long-standing question of evolutionary genetics, which is more generally related to the determination of the adaptive substitution rate and its relationship with species effective population size (N_e) and genetic diversity. Empirical evidence reported so far is equivocal, with some but not all studies supporting a higher adaptive substitution rate in large-N_e than in small-N_e species. We gathered coding sequence polymorphism data and estimated the adaptive amino-acid substitution rate ω_a, in 50 species from ten distant groups of animals with markedly different population mutation rate θ. We reveal the existence of a complex, timescale dependent relationship between species adaptive substitution rate and genetic diversity. We find a positive relationship between ω_a and θ among closely related species, indicating that adaptation is indeed limited by the mutation supply, but this was only true in relatively low-θ taxa. In contrast, we uncover no significant correlation between ω_a and θ at a larger taxonomic scale, suggesting that the proportion of beneficial mutations scales negatively with species' long-term N_e.