DETROIT—The Motor City has lots of drafty houses and tens of thousands of unemployed people. So when Congress proposed spending $5 billion to insulate homes as part of the stimulus bill, Detroit got excited. The director of the city agency managing the program advertised for construction companies before the legislation even passed.

But on the same day in March 2009 that Shenetta Coleman picked up applications from 46 companies, she received an email from the Michigan Department of Human Services telling her she couldn't...