We investigate Holder regularity of adjoint states and optimal controls for a Bolza problem under state constraints. We start by considering any optimal solution satisfying the constrained maximum principle in its normal form and we show that whenever the associated Hamiltonian function is smooth enough and has some monotonicity properties in the directions normal to the constraints, then both the adjoint state and optimal trajectory enjoy Holder type regularity. More precisely, we prove that if the state constraints are smooth, then the adjoint state and the derivative of the optimal trajectory are Holder continuous, while they have the two sided lower Holder continuity property for less regular constraints. Finally, we provide sufficient conditions for Holder type regularity of optimal controls.