The law also allowed states to establish their own goals for improving graduation rates. Many set them low. Nevada, for instance, pledged to get just 50 percent of its students to graduate on time. And since the law required no annual measures of progress, California proposed that even a one-tenth of 1 percent annual improvement in its graduation rate should suffice.

Daniel J. Losen, who has studied dropout reporting for the Civil Rights Project at the University of California, Los Angeles, said he once pointed out to a state official that, at that pace, it would take California 500 years to meet its graduation goal.

“In California, we’re patient,” Mr. Losen recalled the official saying.

Most troublesome to some experts was the way the No Child law’s mandate to bring students to proficiency on tests, coupled with its lack of a requirement that they graduate, created a perverse incentive to push students to drop out. If low-achieving students leave school early, a school’s performance can rise.

No study has documented that the law has produced such an effect nationwide. Experts say they believe many low-scoring students are prodded to leave school, often by school officials urging them to seek an equivalency certificate known as a General Educational Development diploma.

“They get them out so they don’t have them taking those tests,” said Wanda Holly-Stirewalt, director of a program in Jackson, Miss., that helps dropouts earn a G.E.D. “We’ve heard that a lot. It happens all over the system.”

After several research groups questioned graduation rates, the federal Department of Education in 2005 published an estimated rate for each state, to identify those that were reporting least accurately. The figures suggested that nine states had overstated their graduation rates by 10 to 22 percentage points.

Part of the discrepancy is because many states inflate their official rate by counting dropouts who later earn a G.E.D. as graduates or by removing them from calculations altogether.