It is known for a long time that quantum systems can display correlations that can not be explained using classical non-contextual theories. For the Klyachko inequality, the sum of the probabilities of the events considered can be at most 2 when we use these theories, while for quantum mechanics we can reach a value around 2.236. However, general probability theories allow a violation of 2.5. With that in mind, one can ask what physical principle forbids higher violations than the quantum maximum. Here we show that for some scenarios, higher than quantum violations can be ruled out using the exclusivity principle, which can be stated as follows: the sum of pairwise exclusive events cannot exceed 1.