We routinely encounter speakers with different accents and speaking styles. The speech perception literature offers examples of disruption of comprehension for unfamiliar speech and also of listeners’ rapid accommodation to unfamiliar accents. Much of this research uses a single measure and/or focuses on isolated word perception. We investigated listeners’ abilities to comprehend and shadow connected speech spoken in a familiar or unfamiliar accent. We found increases in shadowing latencies and comprehension errors in the Dissimilar Speech relative to Similar Speech conditions—especially for relatively informal rather than more academic style speech. Additionally, there was less accommodation over time to Dissimilar than Similar Speech. These results suggest that there are costs both in the immediate timescale of processing speech (necessary for shadowing) and in the longer time scale of listening comprehension when accent and other speech quality is very different from one’s own speech.