Humans and animals must often discriminate between complex natural sounds in the presence of competing sounds (maskers). Although the auditory cortex is thought to be important in this task, the impact of maskers on cortical discrimination remains poorly understood. We examined neural responses in zebra finch (Taeniopygia guttata) field L (homologous to primary auditory cortex) to target birdsongs that were embedded in three different maskers (broadband noise, modulated noise and birdsong chorus). We found two distinct forms of interference in the neural responses: the addition of spurious spikes occurring primarily during the silent gaps between song syllables and the suppression of informative spikes occurring primarily during the syllables. Both effects systematically degraded neural discrimination as the target intensity decreased relative to that of the masker. The behavioral performance of songbirds degraded in a parallel manner. Our results identify neural interference that could explain the perceptual interference at the heart of the cocktail party problem.