Actually, Western Washington does "run out of rain" during the summers. I believe the region is considered semi-arid or arid during June - September. As local gardeners know, you need to be vigilant about watering your plants in the summer, if they're not of the wet-winter, dry-summer liking types. As many Seattlites like to point out, we get less rainfall on a yearly basis than New York City, but most of it falls during the winter months.
During winter, Western Washington relies on rain for water. In summer, we rely mainly on melting mountain snowpack as the water source. This means that if there is a relatively dry winter with little snow in the mountains, there will be less water available during the summer months, especially when you consider increased evaporation effects and demand during that time. We have reservoirs that we fill up in the winter and spring, and draw on during the dry months, which helps with supply, but this can only help so much.
There's also population growth and climate change effects. An increase in population means and increased demand for water. Climate models have predicted that the weather will become more extreme in the Pacific NW, meaning there would be more floods and more droughts than we currently see.
Seattle has grown a lot over the last 50 years, but the overall use of water has remained steady, which is attributed (mostly or at least in part) to individual water conservation efforts.