For a while, it looked like a federal measure aimed at supporting teachers and other state and local workers had helped save Amanda VanNess's job. It turned out it was only a temporary reprieve.

In July, the 25-year-old Ms. VanNess lost her job teaching kindergarten at Pickett Elementary School in Toledo, Ohio, as a result of cuts that came after voters rejected an income-tax increase aimed at shoring up a budget deficit at the Toledo Public School District.

In...