  The holographic bound, $S<=A/4{\ell^2_P}$, asserts that the entropy $S$ of a
system is bounded from above by a quarter of the area $A$ of a circumscribing
surface measured in Planck areas. This bound is widely regarded as part of the
elusive fundamental theory of nature. In fact, the bound is known to be valid
for generic weakly gravitating isolated systems in {\it three} spatial
dimensions. Nevertheless, the entropy content of a physical system is expected
to be an increasing function of the number of spatial dimensions (the more the
dimensions, the more ways there are to split up a given amount of energy).
Thus, one may expect the challenge to the holographic entropy bound to become
more and more serious as the number of spatial dimensions increases. In this
paper we explicitly show that thermal radiation in $D$ flat spatial dimensions
with $D\gtrsim 10^2$ may indeed violate the holographic entropy bound.
