DONETSK, Ukraine — The warnings from the North Atlantic Treaty Organization and the White House over the past week could not have been graver in tone: The Russian Army, they said, had massed enough forces on the border with Ukraine to invade.

The last time Russian troops appeared to menace Ukraine, in the spring, the Ukrainian military quickly halted attacks on pro-Russian separatists to avoid the chance of touching off a new war in Europe. Not this time.

Buoyed by successes against the separatists over the past two months — and noting that the Russians have threatened an invasion in the region before without following through — Ukrainian commanders have pressed ahead with an offensive to drive the rebels from their stronghold in Donetsk in the east.

The army continued to fire artillery into the city nightly, and paramilitary groups raided outlying villages despite warnings from President Vladimir V. Putin of Russia that he could intervene at any time to protect Ukrainians who favor closer ties with his country. And the Ukrainians have flaunted their victories.