  We consider the Newtonian limit of modified theories of gravity that include
inverse powers of the curvature in the action in order to explain the cosmic
acceleration. It has been shown that the simplest models of this kind are in
conflict with observations at the solar system level. In this letter we point
out that when one adds to the action inverse powers of curvature invariants
that do not vanish for the Schwarzschild geometry one generically recovers an
acceptable Newtonian limit at small distances. Gravity is however modified at
large distances. We compute the first correction to the Newtonian potential in
a quite general class of models. The characteristic distance entering in these
modifications is of the order of 10pc for the Sun and of the order of 10^2 kpc
for a galaxy.
