If it seems like it's snowed a lot this winter, you would be right, alhtough December really skewed things.
Right now, Sea-Tac Airport stands at its 8th snowiest winter since records begain in 1948 at 22.8" -- just one-tenth of an inch behind No. 7. and a half inch behind No. 6.
The unofficial snow season in Seattle runs October 15 to April 15, but usually once we get past mid-March, it's game over. That said, it snowed a half inch in late March and even a trace of snow on April 19th and 20th, so it's not out of the question.
Also, long range forecasts do keep it cooler than normal through the end of the month. I don't see any big arctic blasts or widespread snow events, but enough to maybe sneak in 0.1"? Perhaps.
Here is the Top 10 chart: