  The universal bound on specific entropy was originally inferred from black
hole thermodynamics. We here show from classical thermodynamics alone that for
a system at fixed volume or fixed pressure, the ratio of entropy to
nonrelativistic energy has a unique maximum $(S/E)_\mathrm{max}$. A simple
argument from quantum dynamics allows one to set a model--independent upper
bound on $(S/E)_\mathrm{max}$ which is usually much tighter than the universal
bound. We illustrate with two examples.
