Internally resonant dynamics in the nonlinear free vibrations of suspended cables are analytically investigated by means of a multi-mode Galerkin-based discretization and second-order multiple scales. Emphasis is placed on planar 2:1 internal resonances. The equations of motion of a general inclined cable model, which account for the dynamic extensibility effects and the system asymmetry due to inclined equilibrium, are considered. By considering higher-order effects due to quadratic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations associated with the resonant nonlinear normal modes reveal the dependence of cable nonlinear response on different resonant and non-resonant modes. Based on the modal convergence properties performed on the resonantly activated cables, the illustrative results provide hints for proper reduced-order model selections from the asymptotic solution. The underlying effects of cable inclination and cable sag are presented. The theoretical predictions are validated by finite difference numerical time laws of the original system equations of motion.