We construct a class of counterexamples to cosmic censorship in four dimensional Einstein–Born–Infeld theory with asymptotically anti-de Sitter boundary conditions, and investigate the effect of the Born–Infeld parameter b in comparison with the counterpart of Einstein–Maxwell theory. When a charged massive scalar field is included into the action, we find that this class of counterexamples to cosmic censorship would be removed if the charge of scalar fields is above the bound of charge q w . In particular, the bound of charge required to preserve cosmic censorship increases with the increasing of Born–Infeld parameter. Meanwhile, we also show the lower bounds on charge-to-mass ratio with the different values of Born–Infeld parameter.