We introduce a generalized class of states called K-quantum nonlinear coherent states. Each K-state has K j-components corresponding to one and the same eigenvalue. Each Kj -component can be composed of K K = 1-states in a correlated manner. These states are shown to be realized in the long-term behavior of the vibrational motion of an ion properly trapped and laser-driven. Nonclassical properties of the states are studied in detail.