Sentence comprehension requires inferring, from a sequence of words, the structure of syntactic relationships that bind these words into a semantic representation. Our limited ability to build some specific syntactic structures, such as nested center-embedded clauses (e.g., “The dog that the cat that the mouse bit chased ran away”), suggests a striking capacity limitation of sentence processing, and thus offers a window to understand how the human brain processes sentences. Here, we review the main hypotheses proposed in psycholinguistics to explain such capacity limitation. We then introduce an alternative approach, derived from our recent work on artificial neural networks optimized for language modeling, and predict that capacity limitation derives from the emergence of sparse and feature-specific syntactic units. Unlike psycholinguistic theories, our neural network-based framework provides precise capacity-limit predictions without making any a priori assumptions about the form of the grammar or parser. Finally, we discuss how our framework may clarify the mechanistic underpinning of language processing and its limitations in the human brain.