The history of the Bellevue Art Museum, interestingly, does not begin with a museum. In 1947 a group of volunteers formed the Pacific Northwest Arts and Crafts Association. In July of that year they put on an Arts and Crafts street fair. This fair occurs yearly to this day and has developed into a rather large deal.
The Bellevue Art Museum itself was established in 1975, and was located in an old schoolhouse. Later it moved into a former funeral parlor and then into the third floor of the Bellevue Shopping Center, before finally moving into its current building in 2000.
Designed by Steven Holl, this building was supposed to be a "prototype for urbanizing a sprawling urban zone" through being compact, utilizing underground parking, and having a lot-filling form. I'm not sure how well it achieves this goal, but it is definitely an interesting building. The exterior may be a bit strange, but the interior spaces are pretty awesome, and many are day lit.
In 2003 the museum was forced to close due to funding issues, and did not reopen until mid-2005. Since then it has reinvented itself by rededicating to its crafts roots and focusing heavily on education. It also distinguishes itself by not having a permanent collection.