Performing a cognitive task while maintaining upright stance can lead to increased or reduced body sway depending on tasks and experimental conditions. Because greater sway is commonly taken to indicate loosened postural control, and vice versa, the precise impact of cognitive load on postural stability has remained unclear. In much of the large literature on posture-cognition dual tasking, the assigned postural task is to simply maintain stance (so-called "quiet standing"). This contrasts with quotidian use of postural coordination to maintain balance while also facilitating suprapostural sensorimotor tasks. In this study, healthy young participants either maintained quiet stance or carried out a visuopostural alignment task while performing a spatial, nonspatial, or no additional cognitive task. Body sway increased during both cognitive tasks while quiet standing, as is often observed, but not while performing the visuopostural alignment task. This result is not consistent with the commonly invoked competition between tasks for limited processing resources. It suggests that constraints placed on posture control by suprapostural task goals may significantly alter interactions between posture control and cognitive tasks, and that dual-task results obtained under quiet standing conditions may not generalize to postural control in everyday task conditions.