“I can’t take it anymore,” said Gov. Christine Gregoire, a Democrat. “I sure hope that’s where the people of the State of Washington are.”

So now, facing a budget gap of nearly $2 billion and more bleak revenue forecasts, Ms. Gregoire and some Democratic leaders say they will seek the same solution they have in the past: To raise taxes.

The question is whether, this time, voters will actually let them.

It was barely a year ago that voters rejected nearly $1 billion in increases, largely through sales taxes on things like soda and candy. Now, Ms. Gregoire is being regarded as somewhat daring — and she has struck a more animated public posture — in proposing a temporary half-cent increase in the state sales tax. If passed, it would raise $494 million, filling only about one-fourth of the projected hole. California, by comparison, is considering a slew of tax increase proposals that could easily exceed $10 billion.

Ms. Gregoire is walking a particularly tight political rope. The Legislature is controlled by Democrats, but a ballot measure passed last year requires a two-thirds vote for lawmakers to raise taxes. Washington also has no income tax, and last fall voters resoundingly rejected a well-financed effort, led by Bill Gates Sr., the father of the Microsoft co-founder, to levy one on the wealthy. And some people who might support an income tax on the wealthy, including among the Occupy protesters who showed up in force at the Capitol this week, oppose an increase in the sales tax. The current state sales tax is 6.5 cents, but it is 9.5 percent in Seattle after local surcharges.

“They think it’s regressive, and it is,” Ms. Gregoire said in an interview. “But you know what my line back to them is? ‘Cutting kids so they’re not getting a decent education is regressive. Cutting services for the disabled so they have no place to go is regressive.’ ”