The canonical entropy SK for an ideal (noninteracting) system is compared with the canonical entropy SH for a similar nonideal system at the same temperature. Using the Gibbs-Bogoliubov inequality, upper and lower bounds are derived for (SH−SK). For classical fluids and for quantum or classical spin systems, the upper bound is found to be identically zero, yielding SH⩽SK. The classical fluid result agrees with a recent finding of Baierlein and generalizes it to a class of momentum-dependent potentials. The spin system result further extends the scope of Baierlein's inequality. For quantum fluids, a discussion of the bounds is given, but no conclusive results are obtained.