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Existing speech enhancement algorithms can improve speech quality but not speech intelligibility, and the reasons for that are unclear. In the present paper, we present a theoretical framework that can be used to analyze potential factors that can influence the intelligibility of processed speech. More specifically, this framework focuses on the fine-grain analysis of the distortions introduced by speech enhancement algorithms. It is hypothesized that if these distortions are properly controlled, then large gains in intelligibility can be achieved. To test this hypothesis, intelligibility tests are conducted with human listeners in which we present processed speech with controlled speech distortions. The aim of these tests is to assess the perceptual effect of the various distortions that can be introduced by speech enhancement algorithms on speech intelligibility. Results with three different enhancement algorithms indicated that certain distortions are more detrimental to speech intelligibility degradation than others. When these distortions were properly controlled, however, large gains in intelligibility were obtained by human listeners, even by spectral-subtractive algorithms which are known to degrade speech quality and intelligibility.