The paper deals with the problem of adaptive attenuation of disturbances with uncertain and possibly time-varying characteristics when the plant model is also affected by a non-negligible uncertainty. The proposed solution is based on the Adaptive Switching Control paradigm and relies on a family of pre-synthesized controllers, such that, for any possible operating condition, at least one controller is able to ensure desired attenuation capabilities. Then, a supervisory unit selects the controller providing the best potential performance on the basis of appropriately defined test functionals. We show that, when each of the candidate controllers robustly stabilizes the uncertain plant, it is possible to implement the switching controller so that robust stability is guaranteed for any possible switching sequence. Further, we study the properties of the proposed test functionals by analyzing the effects of the plant/model mismatch on their inference capabilities. Simulation results are shown to validate the proposed control solution.