WASHINGTON—After being buffeted by Republican victories in the 2010 midterm elections, White House aides saw education as ripe for bipartisan cooperation. Both parties wanted to address complaints about the No Child Left Behind law. Congress seemed prepared to act.

But while the White House talked up cooperation in public, in private it was preparing Plan B. In December that year, Education Secretary Arne Duncan warned top Democrats that if Congress didn't act, the administration would use executive authority to essentially...