During listening, brain activity tracks the rhythmic structures of speech signals. Here, we directly dissociated the contribution of neural envelope tracking in the processing of speech acoustic cues from that related to linguistic processing. We examined the neural changes associated with the comprehension of Noise-Vocoded (NV) speech using magnetoencephalography (MEG). Participants listened to NV sentences in a 3-phase training paradigm: (1) pre-training, where NV stimuli were barely comprehended, (2) training with exposure of the original clear version of speech stimulus, and (3) post-training, where the same stimuli gained intelligibility from the training phase. Using this paradigm, we tested if the neural responses of a speech signal was modulated by its intelligibility without any change in its acoustic structure. To test the influence of spectral degradation on neural envelope tracking independently of training, participants listened to two types of NV sentences (4-band and 2-band NV speech), but were only trained to understand 4-band NV speech. Significant changes in neural tracking were observed in the delta range in relation to the acoustic degradation of speech. However, we failed to find a direct effect of intelligibility on the neural tracking of speech envelope in both theta and delta ranges, in both auditory regions-of-interest and whole-brain sensor-space analyses. This suggests that acoustics greatly influence the neural tracking response to speech envelope, and that caution needs to be taken when choosing the control signals for speech-brain tracking analyses, considering that a slight change in acoustic parameters can have strong effects on the neural tracking response.