SOLEDAR, Ukraine—Gen. Oleksandr Rozmaznin was taken aback when his Russian counterpart, assigned to help oversee the cease-fire in eastern Ukraine, announced on Jan. 20 that he would no longer show up at their joint office.

There was too great a security threat, Gen. Alexander Vyaznikov said, pointing to Grad rockets that had fallen nearby the day before, the latest in months of attacks.

“I...