When the Los Angeles Contemporary Museum of Art recently deaccessioned a trove of costumes from its collection, a textile artist stumbled upon an auction catalog. His new project was born when he scooped up more than 50 deaccessioned costumes.

It's easy to walk into any of the myriad free art museums here in Washington and find days' worth of great stuff to see. But what about the chances of you getting your artwork up in one of those museums?…