Purpose: This study investigated how listeners process acoustic cues preserved during sentences interrupted by nonsimultaneous noise that was amplitude modulated by a competing talker. Method: Younger adults with normal hearing and older adults with normal or impaired hearing listened to sentences with consonants or vowels replaced with noise amplitude modulated by a competing talker. Sentences were spectrally shaped according to individual audiograms or to the mean audiogram from the listeners with hearing impairment for a younger spectrally shaped control group. The modulation spectrum of the noise was low-pass filtered at different modulation cutoff frequencies. The effect of noise level was also examined. Results: Performance declined when nonsimultaneous masker modulation included faster rates and was maximized when masker modulation matched the preserved primary speech modulation. Vowels resulted in better performance compared with consonants at slower modulation cutoff rates, likely due to suprasegmental features. Poorer overall performance was observed with increased age or hearing loss, and for listeners who received spectrally shaped speech. Conclusions: Nonsimultaneous amplitude modulations from a competing talker significantly interacted with the preserved speech segment, and additional listener factors were observed for age and hearing loss. Importantly, listeners may obtain benefit from nonsimultaneous competing modulations when they match the preserved modulations of the sentence.