  Correlation boxes are hypothetical systems capable of producing the maximal
algebraic violation of Bell inequalities, beyond the quantum bound and without
superluminal signaling. The fact that these systems show stronger correlations
than those presented by maximally entangled quantum states has been regarded as
a demonstration that the former are more nonlocal than the latter. By employing
an alternative, consistent measure of nonlocality, we show that this conclusion
is not necessarily true. In addition, we find a class of correlation boxes that
are less nonlocal than the quantum singlet with respect to the
Clauser-Horne-Shimony-Holt inequality, being, at the same time, more nonlocal
with respect to the 3322 inequality.
