My wife and I are not the speediest climbers and are a bit worried about having enough time to make it from Colonial campground to Washington pass in one day. If we just poop out, is there anywhere on the way up to get water, or to bivouac for the night if need be?
I don't remember any campgrounds or water sources after Colonial Creek. As Fred pointed out, you might luck out and get water from people in cars. And if you have a water filtration system there are numerous streams you can use. There is a pit toilet between Rainy and Washington Passes, but no water.
As for camping, it could be possible to stealth camp off the road. But two important things: Camping outside of official campgrounds is not permitted in National Parks, and you'll be in North Cascades National Park on the climb. And there will probably be snow on the side of the road in higher elevations, even in mid-June. The snow will be deeper the higher up you go.
To give you an example, my girlfriend April and I crossed Washington/Rainy last year on June 11th. This was two weeks after the passes opened, and it was a very high snowfall winter. We started to see snow on the side of the road around 3,000 feet in elevation. And at either pass the snow was several feet deep.
I'm not saying it will be the same when you cross, but there can still be snow up there!
This was at about 4,000 feet.
Rainy Pass, elevation 4,855 feet. Snow taller than our bikes.