WASHINGTON—U.S. employers added jobs at a robust clip in June and the unemployment rate fell, signs of labor-market strength as the economic recovery heads into its sixth year.

Nonfarm employment advanced at a seasonally adjusted 288,000 last month, the Labor Department said Thursday. The combined gains for the prior two months were revised up 29,000. April's 304,000 increase was the strongest since January 2012.

The...