Speech recognition is diminished when a listener has an auditory temporal processing deficit. Such deficits occur in listeners over 65 years old with normal hearing (NH) and with age-related hearing loss, but their source is still unclear. These deficits may be especially apparent when speech occurs at a rapid rate and when a listener is mostly reliant on temporal information to recognize speech, such as when listening with a cochlear implant (CI) or to vocoded speech (a CI simulation). Assessment of the auditory temporal processing abilities of adults with CIs across a wide range of ages should better reveal central or cognitive sources of age-related deficits with rapid speech because CI stimulation bypasses much of the cochlear encoding that is affected by age-related peripheral hearing loss. This study used time-compressed speech at four different degrees of time compression (0, 20, 40, and 60%) to challenge the auditory temporal processing abilities of younger, middle-aged, and older listeners with CIs or with NH. Listeners with NH were presented vocoded speech at four degrees of spectral resolution (unprocessed, 16, 8, and 4 channels). Results showed an interaction between age and degree of time compression. The reduction in speech recognition associated with faster rates of speech was greater for older adults than younger adults. The performance of the middle-aged listeners was more similar to that of the older listeners than to that of the younger listeners, especially at higher degrees of time compression. A measure of cognitive processing speed did not predict the effects of time compression. These results suggest that central auditory changes related to the aging process are at least partially responsible for the auditory temporal processing deficits seen in older listeners, rather than solely peripheral age-related changes.