Russia has likely been secretly carrying out nuclear tests with very low explosive power to help it upgrade its nuclear arsenal, according to a new U.S. intelligence assessment that challenges Moscow’s claims that it is faithfully adhering to an international treaty banning nuclear tests.

The assessment, which was reported by The Wall Street Journal before being made public Wednesday, marks the first time the U.S. has said the Kremlin hasn’t strictly observed its commitments under the Comprehensive Nuclear Test Ban Treaty....