LUHANSKE, Ukraine—Nine hours after a cease-fire was supposed to take effect, rockets and mortars slammed into Ukrainian army Lt. Yuriy Brekharya’s redoubt outside the besieged city of Debaltseve “like never before,” he said.

Three days of relentless pounding later, he appealed to a neighboring commander for help after advancing Russia-backed separatists destroyed his unit’s final armored vehicle.

“Buddy,...