Recent animal research has shown that exposure to single episodes of intense noise causes cochlear synaptopathy without affecting hearing thresholds. It has been suggested that the same may occur in humans. If so, it is hypothesized that this would result in impaired encoding of sound and lead to difficulties hearing at suprathreshold levels, particularly in challenging listening environments. The primary aim of this study was to investigate the effect of noise exposure on auditory processing, including the perception of speech in noise, in adult humans. A secondary aim was to explore whether musical training might improve some aspects of auditory processing and thus counteract or ameliorate any negative impacts of noise exposure. In a sample of 122 participants (63 female) aged 30–57 years with normal or near-normal hearing thresholds, we conducted audiometric tests, including tympanometry, audiometry, acoustic reflexes, otoacoustic emissions and medial olivocochlear responses. We also assessed temporal and spectral processing, by determining thresholds for detection of amplitude modulation and temporal fine structure. We assessed speech-in-noise perception, and conducted tests of attention, memory and sentence closure. We also calculated participants' accumulated lifetime noise exposure and administered questionnaires to assess self-reported listening difficulty and musical training. The results showed no clear link between participants' lifetime noise exposure and performance on any of the auditory processing or speech-in-noise tasks. Musical training was associated with better performance on the auditory processing tasks, but not the on the speech-in-noise perception tasks. The results indicate that sentence closure skills, working memory, attention, extended high frequency hearing thresholds and medial olivocochlear suppression strength are important factors that are related to the ability to process speech in noise.