Grammatical-specific language impairment (G-SLI) in children, arguably, provides evidence for the existence of a specialised grammatical sub-system in the brain, necessary for normal language development. Some researchers challenge this, claiming that domain-general, low-level auditory deficits, particular to rapid processing, cause phonological deficits and thereby SLI. We investigate this possibility by testing the auditory discrimination abilities of G-SLI children for speech and non-speech sounds, at varying presentation rates, and controlling for the effects of age and language on performance. For non-speech formant transitions, 69% of the G-SLI children showed normal auditory processing, whereas for the same acoustic information in speech, only 31% did so. For rapidly presented tones, 46% of the G-SLI children performed normally. Auditory performance with speech and non-speech sounds differentiated the G-SLI children from their age-matched controls, whereas speed of processing did not. The G-SLI children evinced no relationship between their auditory and phonological/grammatical abilities. We found no consistent evidence that a deficit in processing rapid acoustic information causes or maintains G-SLI. The findings, from at least those G-SLI children who do not exhibit any auditory deficits, provide further evidence supporting the existence of a primary domain-specific deficit underlying G-SLI.