People with hearing impairment are thought to rely heavily on context to compensate for reduced audibility. Here, we explore the resulting cost of this compensatory behavior, in terms of effort and the efficiency of ongoing predictive language processing. The listening task featured predictable or unpredictable sentences, and participants included people with cochlear implants as well as people with normal hearing who heard full-spectrum/unprocessed or vocoded speech. The crucial metric was the growth of the pupillary response and the reduction of this response for predictable versus unpredictable sentences, which would suggest reduced cognitive load resulting from predictive processing. Semantic context led to rapid reduction of listening effort for people with normal hearing; the reductions were observed well before the offset of the stimuli. Effort reduction was slightly delayed for people with cochlear implants and considerably more delayed for normal-hearing listeners exposed to spectrally degraded noise-vocoded signals; this pattern of results was maintained even when intelligibility was perfect. Results suggest that speed of sentence processing can still be disrupted, and exertion of effort can be elevated, even when intelligibility remains high. We discuss implications for experimental and clinical assessment of speech recognition, in which good performance can arise because of cognitive processes that occur after a stimulus, during a period of silence. Because silent gaps are not common in continuous flowing speech, the cognitive/linguistic restorative processes observed after sentences in such studies might not be available to listeners in everyday conversations, meaning that speech recognition in conventional tests might overestimate sentence-processing capability.