The paper establishes the basic convergence conditions for the method of nonconforming finite elements applied to a class of generalized elliptic boundary value problems with variable, not necessarily smooth coefficients. The main result is a new, generalized patch test. Approximability and success in this test is the necessary and sufficient condition for convergence of the nonconforming approximations. It is proved that nonconforming elements of Wilson, Adini, Crouzeit–Raviart, Morley, and de Veubeke pass the generalized patch test and thus yield convergent approximations of the boundary value problems.