In the Palatini version of f(R) theories of gravity, the independent connection (that in principle can be both nonmetric and nonsymmetric) can always be algebraically eliminated in favor of the metric and the matter fields, so long as it is not coupled to the matter explicitly. I will show it is a special characteristic of f(R) actions and it is not true for actions that include other curvature invariants. I will also consider metric-affine theories of gravity, that is theories described by actions including covariant derivatives of the matter fields. I will show that the connection is non-dynamical for the most general action containing second order invariants of the curvature and the torsion. However, including higher order terms excites new degrees of freedom, making the connection (or parts of it) dynamical.