In this paper we discuss a novel route to chaos in cam-follower systems, a class of mechanical devices of relevance in applications. We show that a sudden transition to chaos is observed when a complete chattering motion (characterized theoretically by infinitely many impacts) is interrupted. Using numerical computations and semi-analytical methods we show that a cascade of grazing bifurcations is then originated, leading the system into chaos. This novel phenomenon is described using an appropriate model that is shown to predict correctly the numerical observations. The results are used to explain the transition to chaos observed in an experimental cam-follower system recently presented in the literature.