The number of measles cases in the U.S. this year has hit its highest level in more than 25 years, the Centers for Disease Control and Prevention reported, warning that a major public-health milestone—the elimination of measles in the U.S.—may be sacrificed if two large outbreaks aren’t curbed by the fall.

The CDC said 971 cases have been reported so far in 2019, including 91 cases in the past week and a half. The tally means that in just five months, the outbreaks have surpassed the 963 cases reported in all of 1994, the...