  Quantum correlations resulting in violations of Bell inequalities have
generated a lot of interest in quantum information science and fundamental
physics. In this paper, we address some questions that become relevant in
Bell-type tests involving systems with local dimension greater than 2. For
CHSH-Bell tests within 2-dimensional subspaces of such high-dimensional
systems, it has been suggested that experimental violation of Tsirelson's bound
indicates that more than 2-dimensional entanglement was present. We explain
that the overstepping of Tsirelson's bound is due to violation of fair
sampling, and can in general be reproduced by a separable state, if fair
sampling is violated. For a class of Bell-type inequalities generalized to
d-dimensional systems, we then consider what level of violation is required to
guarantee d-dimensional entanglement of the tested state, when fair sampling is
satisfied. We find that this can be used as an experimentally feasible test of
d-dimensional entanglement for up to quite high values of d.
