  Quantum correlations which violate a Bell inequality are presumed to power
better-than-classical protocols for solving communication complexity problems
(CCPs). How general is this statement? We show that violations of
correlation-type Bell inequalities allow advantages in CCPs, when communication
protocols are tailored to emulate the Bell no-signaling constraint (by not
communicating measurement settings). Abandonment of this restriction on
classical models allows us to disprove the main result of, inter alia, [Brukner
et. al., Phys Rev. Lett. 89, 197901 (2002)]; we show that quantum correlations
obtained from these communication strategies assisted by a small quantum
violation of the CGLMP Bell inequalities do not imply advantages in any CCP in
the input/output scenario considered in the reference. More generally, we show
that there exists quantum correlations, with nontrivial local marginal
probabilities, which violate the $I_{3322}$ Bell inequality, but do not enable
a quantum advantange in any CCP, regardless of the communication strategy
employed in the quantum protocol, for a scenario with a fixed number of inputs
and outputs
