In this paper we investigate some Nystr\"om methods for Fredholm integral equations in the interval [0,1]. We give an overview of the order of convergence, which depends on the smoothness of the involved functions. In particular, we consider the Nystr\"om methods based on the so called Generalized Bernstein quadrature rule, on a Romberg scheme and on the so-called IMT rule. We prove that the proposed methods are convergent, stable and well conditioned. Also, we give several numerical tests for comparing these three methods.