The attentional blink (AB) is a widely studied deficit in reporting the second of two sequentially presented targets when they occur within 500 milliseconds. The AB often is interpreted to index a structural limit in sequential visual processing. However, this interpretation is challenged by reports that the deficit can be reduced with several hundred trials of specific training (Braun in Nature, 393(6684), 424–425, 1998; Choi et al. in Proceedings of the National Academy of Sciences, 109(30), 12242–12247, 2012; Taatgen et al. in Cognitive Psychology, 59(1), 1–29, 2009) and other reports that some individuals experience very little or no deficit, even without specific training (Martens et al. in Journal of Cognitive Neuroscience, 18(9), 1423-1438, 2006). Yet neither of these claims has been studied when the artifact of ceiling effects has been removed. We sent a small number of participants (n = 5) home to practice an AB task on their mobile phones for 3,000-6,000 trials (Experiment 1) and trained a much larger number of participants (n = 48) in a similar way for 1,200-1,800 trials (Experiment 2). Both experiments used adaptive procedures to equate task difficulty throughout training to keep second-target accuracy below ceiling levels. The results showed strong training effects on the rate of processing sequential information. Despite this, there were (a) robust AB effects after training for most participants, (b) no benefit for training on difficult versus easy target tasks, and (c) substantial correlations between the magnitude of the AB before and after extensive training. These findings support the interpretation that the AB is an index of a structural limit in the ability to consciously process rapid visual sequences.