Yes- for the most part they try to go west to east or south to north. There are exceptions, particularly along the I-5 corridor, but usually SR 121 will be west of SR 122 and SR 270 will be south of SR 271 etc.
One digit routes usually get to use up to 99 numbers. SRs 3, 4,6, 7, 8, 9, and I-5 can all go from SR x00 to x99. The exception for that is US-2, which only gets 200 to 209 for use.
Two digit routes get 9 numbers for use, so SR 18 would have SR 181 to SR 189 available or I-90 has SR 900 to SR 909 available.
There are some exceptions:
For numbering purposes, US-101 is SR 1 and SR 11 (since SR 11 is unlikely to spawn a branch and could arguably be a 5xx), so it gets from SR 100 to SR 119. SR 19 is also a functional child of US-101. SR 19 doesn't fit the grid at all (it should be somewhere near the Coulee Dam), and was likely assigned because when they reshuffled highways in 1992 (when SR 19 was assigned), all 10x and 11x numbers but 118 were either in use , being assigned, or being decommissioned and I guess they didn't want to reuse a number or 118 for some reason.
SR 20 actually gets SR 21x numbers since the 20x numbers are used by US-2 and SR 21 doesn't have any branch routes.
SR 16x routes are branches of either SR 16 or SR 410, which is a three digit primary route that can't have its own numbers. Only SR 160, 163, and 166 connect to SR 16. The rest are branches off of SR 410.
US-97 originally got SR 15x numbers (as it fits as SR 15 on the grid), but later routes have been SR 97x.
Same with US-395- it originally got 29x numbers (as it sort of fits as SR 29 on the grid), but later routes have been 39x.
US-195 gets to use 19x numbers since there is no SR 19.
SR 4 actually has domain over the SR 43x series, since SR 4 used to be US-830 and those routes were originally numbered 831, 832, and 833. When they brought SR 4 out, they just went ahead and called them 431, 432, 433.
On a similar note, before it was decommissioned SR 237 was the property of SR 20- it didn't make any numbering sense but the route had been SR 537 until it lost connection with what was then SR 536, and the number had to be changed. Washington used to be very good about being anal about number changes when that sort of situation happened, but have slacked a bit since the reshuffling of 1992 (SR 121, for instance, should now be a 5xx)