Abstract. We prove that if two nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. This naturally gives rise to a special case of a dynamical analogue of the Mordell-Lang conjecture, one that holds for lines in the affine plane A^1 x A^1, under the action of polynomials acting on each coordinate. The proof uses classical results of Ritt for polynomials along with a result of Bilu and Tichy on integral points to establish the result over number fields. The general case over C is then obtained by a specialization argument.