WASHINGTON — President Biden renewed his warning to President Vladimir V. Putin on Tuesday that it would be an “incredibly serious mistake” to use a tactical nuclear weapon in Ukraine, reflecting the increasingly urgent concern in Washington and among Western allies that Russia may be searching for a pretext to unleash such a weapon.“I’m not guaranteeing you that it’s a false flag operation yet, don’t know, but it would be a serious, serious mistake.” Mr. Biden’s comments, and a complex exchange of accusations between Russia and Ukraine, were the latest demonstration of how high tensions are running at a moment when Russia is losing ground and sees no pathway to controlling vast swaths of the four provinces it recently annexed.It is part of what American officials have called an inescapable paradox of the conflict: While the United States and its NATO allies are committed to helping Ukraine expel Russia from its territory, the more successful the Ukrainians are, the greater the risk that Russia will break the unwritten taboo against employing nuclear weapons.Russia’s top military officials began the series of accusations about dirty bombs over the weekend, warning their American counterparts, Defense Secretary Lloyd J. Austin III and the chairman of the Joint Chiefs of staff, Gen. Mark A. Milley, that Ukraine was planning to detonate a radiological attack on Ukrainian soil.Over the past few weeks, the Biden administration has been conducting tabletop exercises, trying to game out how Russia might try to gain an advantage by threatening to use a nuclear weapon — and under what conditions it might actually detonate one."