Several working memory (WM) theories assume a resource sharing between the maintenance of information and its processing, whereas other theories suppose that these two functions of WM rely on different pools of resources. Studies that addressed this question by examining whether dual-task costs occur in tasks combining processing and storage have led to mixed results. Whereas some of them reported symmetric dual-task costs, others found no or negligible effects, while still others found a reduction in performance in memory but not in processing. In the present experiment, we tested whether these discrepancies in results might be due to participants strategically prioritizing one component of the task over the other. Thus, we asked participants to perform at their maximum level (i.e. span level) in one component of the dual task and assessed performance on the other. In line with resourcesharing views, results indicated that performing at span on one task strongly degraded performance on the other, with symmetric costs. However, important residuals in both processing and storage suggested an unexpected resilience of the cognitive system that any resource-sharing theory must take into account.