  We explore the challenges posed by the violation of Bell-like inequalities by
$d$-dimensional systems exposed to imperfect state-preparation and measurement
settings. We address, in particular, the limit of high-dimensional systems,
naturally arising when exploring the quantum-to-classical transition. We show
that, although suitable Bell inequalities can be violated, in principle, for
any dimension of given subsystems, it is in practice increasingly challenging
to detect such violations, even if the system is prepared in a maximally
entangled state. We characterize the effects of random perturbations on the
state or on the measurement settings, also quantifying the efforts needed to
certify the possible violations in case of complete ignorance on the system
state at hand.
