<p>We study the multiplicity of solutions for nonlinear Hamiltonian systems of Schrödinger equations with concave-convex nonlinearities. Firstly, when the convex nonlinearity exhibits the Sobolev subcritical growth, we establish the existence of two distinct sequences of solutions: one with unbounded energy and the other with negative energy converging to zero. Our approach relies on a critical point theorem for strongly indefinite functionals and the Clark theorem. Secondly, when the convex nonlinearity is super-quadratic and possibly Sobolev supercritical, the energy functional is not well defined. To overcome this, we introduce an alternative working space, enabling variational methods to be applied. In this setting, we obtain a sequence of solutions with negative energy approaching zero and vanishing norm.</p>