Standard projective measurements represent a subset of all possible measurements in quantum physics. In fact, non-projective measurements are relevant for many applications, e.g. for estimation problems or transformations among entangled states. In this work we study what quantum measurements can be simulated by using only projective measurements and classical randomness. We first prove that every measurement on a given quantum system can be realised by a projective-simulable measurement on a system enlarged by an ancilla of the same dimension. Then, given a general measurement in dimension two or three, we show that deciding whether it is projective-simulable can be solved by means of semi-definite programming. We also establish conditions for the simulation of measurements using projective ones valid for any dimension. As an application of our formalism, we improve the range of visibilities for which two-qubit Werner states do not violate any Bell inequality for all measurements. From an implementation point of view, our results provide bounds on the amount of noise a general measurement tolerates before losing any advantage over projective ones.