The best predictor of successful language therapy after stroke is a high intensity of treatment (with 5-10 h training per week). However, the necessity of several hours of language exercises each day draws considerably on attentional and cognitive resources of the patients. Thus, not all aphasic patients may be equally suited for intense training approaches. In the present review non-verbal cognitive deficits that often accompany a stroke-induced aphasia are described. Furthermore, initial empirical data on cognitive functions, which predict the success of therapy (intense) after stroke, are summarized. Patients in the acute stage benefit most from intense aphasia treatment, when long-term memory consolidation is relatively preserved. For the chronic stage, indirect evidence suggests that premorbid intelligence as well as attentional functions have positive effects on the success of intense therapy. An empirically based allocation of patients to intense aphasia treatment awaits the results of multicenter trials with sufficiently large sample sizes.