Information unraveling is an elegant theoretical argument suggesting that private information is voluntarily and fully revealed in many circumstances. However, the experimental literature has documented many cases of incomplete unraveling and has suggested limited depth of reasoning on the part of senders as a behavioral explanation. To test this explanation, we modify the design of existing unraveling games along two dimensions. In contrast to the baseline setting with simultaneous moves, we introduce a variant where decision-making is essentially sequential. Second, we vary the cost of disclosure, resulting in a 2×2 treatment design. Both sequential decision-making and low disclosure costs are suitable for reducing the demands on subjects' level-k reasoning. The data confirm that sequential decision-making and low disclosure costs lead to more disclosure, and there is virtually full disclosure in the treatment that combines both. A calibrated level-k model makes quantitative predictions, including precise treatment level and player-specific revelation rates, and these predictions organize the data well. The timing of decisions provides further insights into the treatment-specific unraveling process.