  Why do correlations between the results of measurements performed on physical
systems violate Bell and non-contextuality inequalities up to some specific
limits? The answer may follow from the observation that in quantum theory,
unlike in other theories, whenever there is an experiment to measure $A$
simultaneously with $B$, another to measure $B$ with $C$, and another to
measure $A$ with $C$, there is always an experiment to measure all of them
simultaneously. This property implies that quantum theory satisfies a seemingly
irrelevant restriction called the exclusivity (E) principle which,
surprisingly, explains the set of quantum correlations in some fundamental
scenarios. An open problem is whether the E principle explains the maximum
quantum violation of the Bell-CHSH inequality. Here we show experimentally that
the E principle imposes an upper bound to the violation of the Bell-CHSH
inequality that matches the maximum predicted by quantum theory. For that, we
use the result of an independent experiment testing a specific
non-contextuality inequality. We perform both experiments: the Bell-CHSH
inequality experiment on polarization entangled states of pairs of photons in
Stockholm and, to demonstrate independence, the non-contextuality inequality
experiment on single photons' orbital angular momentum states in Rome. The
observed results provide the first experimental evidence that the E principle
determines the limits of quantum correlations and prove that hypothetical
super-quantum violations for either experiment would violate the E principle.
This supports the conclusion that the E principle captures a fundamental
limitation of nature. If this is true, much of quantum theory trivially follow
from merely taking the E principle to be a fundamental truth, and various
information-theoretic postulates are also simplified and/or strengthened.
