When we grow older, understanding speech in noise becomes more challenging. Research has demonstrated the role of auditory temporal and cognitive deficits in these age-related speech-in-noise difficulties. To better understand the underlying neural mechanisms, we recruited young, middle-aged, and older normal-hearing adults and investigated the interplay between speech understanding, cognition, and neural tracking of the speech envelope using electroencephalography. The stimuli consisted of natural speech masked by speech-weighted noise or a competing talker and were presented at several subject-specific speech understanding levels. In addition to running speech, we recorded auditory steady-state responses at low modulation frequencies to assess the effect of age on nonspeech sounds. The results show that healthy aging resulted in a supralinear increase in the speech reception threshold, i.e., worse speech understanding, most pronounced for the competing talker. Similarly, advancing age was associated with a supralinear increase in envelope tracking, with a pronounced enhancement for older adults. Additionally, envelope tracking was found to increase with speech understanding, most apparent for older adults. Because we found that worse cognitive scores were associated with enhanced envelope tracking, our results support the hypothesis that enhanced envelope tracking in older adults is the result of a higher activation of brain regions for processing speech, compared with younger adults. From a cognitive perspective, this could reflect the inefficient use of cognitive resources, often observed in behavioral studies. Interestingly, the opposite effect of age was found for auditory steady-state responses, suggesting a complex interplay of different neural mechanisms with advancing age. We measured neural tracking of the speech envelope across the adult lifespan and found a supralinear increase in envelope tracking with age. Using a more ecologically valid approach than auditory steady-state responses, we found that young and older, as well as middle-aged, normal-hearing adults showed an increase in envelope tracking with increasing speech understanding and that this association is stronger for older adults.