We present a novel method to compute componentwise ultimate bounds and invariant regions for a class of switching discrete-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The method has the important advantage that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a standard norm for bounding either the perturbation or state vectors, and thus may avoid conservativeness due to different perturbation or state vector components having substantially different bounds. We also establish the relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function. We illustrate the application of our method via numerical examples, including the fault tolerance analysis of the feedback control of a winding machine.