A drastic increase this year in the number of students sitting out New York’s standardized exams has created another kind of test, one for state and federal education officials who must decide whether to punish school districts with low participation rates.

And it is far from certain that they will. Last spring, as anti-testing activists and teachers’ unions rallied parents to have their children opt out, the state and federal Education Departments repeatedly warned that districts with high refusal rates risked losing federal funds. But activists derided these as empty threats, predicting that neither the state nor the federal government would have the stomach to impose penalties in districts where parents were already angry about testing.

Under the Elementary and Secondary Education Act, the federal government has the right to impose various sanctions on states that fail to ensure all students are tested, including withholding Title I funds, which go to schools based on their numbers of poor students. Under the same law, the state itself can withhold funds from districts or schools that do not have sufficient numbers of students tested. The federal government has never imposed such a punishment.

However, no state has had as large an opt-out movement as New York did this year. On Wednesday, the state’s Education Department said that 20 percent of third- through eighth-grade students did not take this year’s exams, quadruple the number from the year before.