Amodal completion refers to the perceptual “filling-in” of partly occluded object fragments. Previous work has shown that object completion occurs efficiently, at early perceptual stages of processing. However, despite efficient early completion, at a later stage, the maintenance of complete-object representations in visual working memory (VWM) may be severely restricted due to limited mnemonic resources being available. To examine for such a limitation, we investigated whether the structure of to-be-remembered objects influences what is encoded and maintained in VWM using a change detection paradigm. Participants were presented with a memory display that contained either “composite” objects, that is, notched shapes abutting an occluding square, or equivalent unoccluded, “simple” objects. The results showed overall increased memory performance for simple relative to composite objects. Moreover, evidence for completion in VWM was found for composite objects that were interpreted as globally completed wholes, relative to local completions or an uncompleted mosaic (baseline) condition. This global completion advantage was obtained only when the “context” of simple objects also supported a global object interpretation. Finally, with an increase in memory set size, the global object advantage decreased substantially. These findings indicate that processes of amodal completion influence VWM performance until some overall-capacity limitation prevents completion. VWM completion processes do not operate automatically; rather, the representation format is determined top-down based on the simple object context provided. Overall, these findings support the notion of VWM as a capacity-limited resource, with storage capacity depending on the structured representation of to-be-remembered objects.