This paper discusses the state bounding problem of switched homogeneous cooperative time-varying systems of degree one. A homogeneous cooperative system is a specific type of nonlinear positive system. For analysis of time-varying systems, we don't use the comparison principle. First, a criterion is presented for the existence of a neighborhood of the equilibrium point where the trajectories of the switched nonlinear positive time-varying system with disturbance asymptotically converge within, and the decay rate of every subsystem is also given. Then, the criteria are extended to switched linear positive time-varying systems. Finally, a numerical example is proposed to illustrate the results.