We report an experiment investigating the “special-process” theory of insight problem solving, which claims that insight arises from non-conscious, non-reportable processes that enable problem re-structuring. We predicted that reducing opportunities for speech-based processing during insight problem solving should permit special processes to function more effectively, thereby facilitating insight. We distracted speech-based processing by using either articulatory suppression or irrelevant speech, with findings for these conditions supporting the predicted insight facilitation effect relative to silent working or thinking aloud. The latter condition was included to investigate the currently contested effect of “verbal overshadowing” on insight, whereby thinking aloud is claimed to hinder the operation of special, non-reportable processes. Whilst verbal overshadowing was not evident in final solution rates, there was nevertheless support for verbal overshadowing up to and beyond the mid-point of the available problem solving time. Overall our data support a special-process theory of insight, whilst also pointing to the role of moderator variables (e.g., available time for solution) in determining the presence or absence of effects predicted by the special process account.