Recent studies of the occurrence of post-flutter limit cycle oscillations (LCO) of the F-16 have provided good support to the long-standing hypothesis that this phenomenon involves a nonlinear structural damping. The proposed mechanism for the appearance of nonlinearity in the damping are the nonlinear geometric effects that arise when the deformations become large enough to exceed the linear regime. A pair of earlier investigations have first developed a finite element-based reduced order modeling (ROM) framework with nonlinearity in damping and potentially nonlinearity in stiffness to predict the aircraft response. These studies have also demonstrated, with an appropriate calibration of the single model parameter, a good match between flight test LCO amplitudes and those predicted by the model. The focus of the present investigation is first to extend the validation of the approach to an interesting set of flight test results shedding important light on the importance of the aerodynamic forces generated by the control surfaces of tip missiles of the F-16 LCO characteristics. The second objective is to include the nonlinearity in stiffness associated with the large deformations that had not been considered before and to assess whether it can be a LCO bounding mechanism.