Clematis wilt is due to one or two fungi that cause leaf and stem lesions, leading to the wilt and dieback. There are few effective preventative controls. However, because the roots are unaffected, plants usually recover. Simply trim out the affected stems.
Often the problem becomes less severe as plants get older. Also, small-flowered types of clematis seem to be less susceptible to the problem. For more on this disease, check this bulletin from the Scott Arboretum at Swarthmore College: http://blogs.scottarboretum.org/gardenseeds/2009/06/clematis-wilt/