  Bekenstein has presented evidence for the existence of a universal upper
bound of magnitude $2\pi R/\hbar c$ to the entropy-to-energy ratio $S/E$ of an
arbitrary {\it three} dimensional system of proper radius $R$ and negligible
self-gravity. In this paper we derive a generalized upper bound on the
entropy-to-energy ratio of a $(D+1)$-dimensional system. We consider a box full
of entropy lowered towards and then dropped into a $(D+1)$-dimensional black
hole in equilibrium with thermal radiation. In the canonical case of three
spatial dimensions, it was previously established that due to quantum buoyancy
effects the box floats at some neutral point very close to the horizon. We find
here that the significance of quantum buoyancy increases dramatically with the
number $D$ of spatial dimensions. In particular, we find that the neutral
(floating) point of the box lies near the horizon only if its length $b$ is
large enough such that $b/b_C>F(D)$, where $b_C$ is the Compton length of the
body and $F(D)\sim D^{D/2}\gg1$ for $D\gg1$. A consequence is that quantum
buoyancy severely restricts our ability to deduce the universal entropy bound
from the generalized second law of thermodynamics in higher-dimensional
spacetimes with $D\gg1$. Nevertheless, we find that the universal entropy bound
is always a sufficient condition for operation of the generalized second law in
this type of gedanken experiments.
