Speech comprehension is preserved up to a threefold acceleration, but deteriorates rapidly at higher speeds. Current models posit that perceptual resilience to accelerated speech is limited by the brain's ability to parse speech into syllabic units using δ/θ oscillations. Here, we investigated whether the involvement of neuronal oscillations in processing accelerated speech also relates to their scale-free amplitude modulation as indexed by the strength of long-range temporal correlations (LRTC). We recorded MEG while 24 human subjects (12 females) listened to radio news uttered at different comprehensible rates, at a mostly unintelligible rate and at this same speed interleaved with silence gaps. δ, θ, and low-γ oscillations followed the nonlinear variation of comprehension, with LRTC rising only at the highest speed. In contrast, increasing the rate was associated with a monotonic increase in LRTC in high-γ activity. When intelligibility was restored with the insertion of silence gaps, LRTC in the δ, θ, and low-γ oscillations resumed the low levels observed for intelligible speech. Remarkably, the lower the individual subject scaling exponents of δ/θ oscillations, the greater the comprehension of the fastest speech rate. Moreover, the strength of LRTC of the speech envelope decreased at the maximal rate, suggesting an inverse relationship with the LRTC of brain dynamics when comprehension halts. Our findings show that scale-free amplitude modulation of cortical oscillations and speech signals are tightly coupled to speech uptake capacity.SIGNIFICANCE STATEMENT One may read this statement in 20-30 s, but reading it in less than five leaves us clueless. Our minds limit how much information we grasp in an instant. Understanding the neural constraints on our capacity for sensory uptake is a fundamental question in neuroscience. Here, MEG was used to investigate neuronal activity while subjects listened to radio news played faster and faster until becoming unintelligible. We found that speech comprehension is related to the scale-free dynamics of δ and θ bands, whereas this property in high-γ fluctuations mirrors speech rate. We propose that successful speech processing imposes constraints on the self-organization of synchronous cell assemblies and their scale-free dynamics adjusts to the temporal properties of spoken language.