In this paper we propose a new technique to obtain upper and lower bounds on the energy norm of the error in the velocity field, for the Stokes problem. It relies on a splitting of the velocity error in two contributions: a projection error, that quantifies the distance of the computed solution to the space of divergence free functions, and an error in satisfying the momentum equation. We will show that both terms can be sharply estimated, from above and from below, by implicit a posteriori error estimators. In particular, the proposed estimator is based on the solution of local Stokes problems both with “Neumann-type” boundary conditions, extending the ideas presented in [12, 17] for the Laplace equation, and homogeneous Dirichlet boundary conditions. The numerical results show very good effectivity indices. The underlying idea is quite general and can be applied to other saddle point problems as well, as the ones arising in mixed formulations of second order PDEs.