In California, storms over the winter break destroyed a million coronavirus test kits that were meant to help schools screen returning students. In Seattle schools, children waited for hours for virus testing, some in a driving rain. In Florida this month, an attempt to supply tests to teachers in Broward County turned up expired kits.

And in Chicago, a labor dispute, partly over testing, kept students out of school for a week.

As millions of American students head back to their desks — Los Angeles, the nation’s second-largest school district, started classes on Tuesday — the coronavirus testing that was supposed to help keep classrooms open safely is itself being tested. In much of the country, things are not going well.