A complete study of non-Markovian effects induced by correlated noise applied to a bistable dynamical system is presented. Starting from the exact functional integral solution of the stochastic equation, it is possible to show that the customary expansion in powers of the characteristic correlation time gives wrong asymptotic results. Other approaches based on a Fokker-Planck equation with a modified diffusion coefficient also fail in reproducing the right long-time behavior of the system. Using a generalized version of instanton calculus of functional integrals, explicit expressions of the invariant measure and transition time between stable fixed points are obtained, in the limit of small noise intensity but arbitrary correlation time. In particular, an original method for extracting the collective degrees of motion has been developed. These analytical results fit, for a large range of parameters, with numerical calculations, giving confidence in the formalism employed.