One possible catalogue number for this object would be E877, which was identified as Athabaskan from Lapierre House, Canada.
These snowshoes seem to be one of two pairs (the upper pair in the case) shown in a circa 1880s photo of an exhibit case/exhibits in the U.S. National Museum (now the Arts and Industries Building). See Smithsonian Institution Archives, Acc. 11-007, Box 020, Image No. MNH-4476 or 4476. See also Record Unit 95, Box 42, Folder 25. http://collections.si.edu/search/detail/edanmdm:siris_arc_391788 . The snowshoes in the case are labelled as from "Indians of Canada."