You’ve seen this movie before: Bitter rivals, bound by the interests of shared history and common geography, find themselves unable to resolve a decades-old policy dispute with potentially world-historical implications. A deadline looms, offering an opportunity for both sides to draw lines in the sand. A pro forma bit of bureaucratic maneuvering escalates into a crisis. Terms are negotiated, but each time a deal seems close it comes unglued at the last minute, for reasons that make more sense to the people involved than to the increasingly anxious and exhausted spectators watching at home—sometimes, it turns out, for reasons as petty as a nasty or unreturned phone call. Frustrated, the leaders turn to the public. One side claims to be on the side of of justice and liberal democracy; the other side appeals to the equally resonant tenets of self-determination and liberty.

Welcome to Jerusalem on the Potomac, or, if you like, Ramallah on the Hill, in which the political leaders of the fiscally challenged United States are playing roles long ago made famous by their stymied counterparts in the Middle East. In the last few days, the fraught negotiations over raising the federal debt ceiling—which are really negotiations about what the government should provide for its citizens—have increasingly come to look and sound like nothing so much as the familiar, tedious peace process. “Out of many, we are one,” President Barack Obama reminded the American people after the collapse of yet another round of talks Monday night. “Peace would herald a new day for both our peoples,” Israeli Prime Minister Benjamin Netanyahu announced before him, in May.

And the debt-ceiling negotiations in Washington are failing for exactly the same reason the Israeli-Palestinian peace negotiations have historically failed, at Camp David, Wye River, and myriad other retreats. The underlying differences—mutually exclusive claims to a finite piece of land in one case, mutually exclusive views of what government is for on the other—are arguably unbridgeable, even in the best of circumstances. As a result, the incentives for those in charge point toward minimizing personal losses over risking career suicide in service of achieving a sweeping solution that is politically risky and possibly untenable. In both situations the losers are the anonymous masses—Social Security recipients, mortgage-seekers, Israelis and Palestinians trying to live normal lives in Sderot or Gaza—but everyone knows who they would blame for a deal they don’t like, whether it’s Obama or Boehner, Netanyahu or Abbas.

Obama and Netanyahu have sought to cast themselves as the responsible adults in their respective rooms—the ones pushing harder, and risking more, to achieve the impossible. “I stood before my people, and I said, ‘I will accept a Palestinian state,’ ” Netanyahu said in that May address, before the U.S. Congress. “I’m willing to take the responsibility,” Obama said last Friday, in a hastily called press conference after Boehner broke off another round of talks. “That’s my job.”

And Obama could fairly be accused of making the same mistake in the debt negotiations that he has in his own efforts in the Middle East: moving too fast to concede ground in the center before reassuring his base that he won’t bend on their red lines, namely Israel’s security and primacy over mercurial Arab allies, or the principles enshrined in the New Deal and the Great Society and everything else the Democrats have spent the last 80 years building.

Nevertheless, the president’s real problem isn’t the fundamentalists on his own side: Nancy Pelosi is no Avigdor Lieberman, despite last week’s grumbling from the Democratic caucus. Just as Hamas holds the ultimate veto over any peace plan Abbas and Netanyahu could bring themselves to agree to, it’s the extremists in the Republican camp who are holding everyone else hostage to their demands. Like militants on the Palestinian side, these angry Republican ideologues have shown themselves constitutionally unwilling to recognize the legitimacy of the other side’s worldview, and perfectly happy to take down the whole ship rather than compromise their ideals.

It’s not surprising that, after last week’s breakdown of so-called grand bargain negotiations between Obama and Boehner, a conspiracy theory began making its way around Washington: that the big blowup was staged to give Boehner some plausible deniability that will allow him to ultimately make a deal. “‘It’ll help if we look mad at each other,’” Politico writer Mike Allen summarized Boehner’s supposed pitch. “If that’s the deal,” he added, “they should both get Academy Awards.” The episode echoed the constant refrain of Israeli diplomats: that the Fatah-aligned leaders of the Palestinian Authority are happy to talk about unity with Hamas in public, while in private they beg the Israelis for help controlling their sometime mortal enemies in Gaza.

So, now we’ve arrived at a place where, it appears, all parties prefer to remain in the limbo of no deal rather than agreeing to a deal they don’t really like, hoping that they might get a less bad deal, or at least make their opponents look worse, tomorrow. Just as the Israelis and the Palestinians perennially look to the Americans or the Europeans to come in and referee, the players in Washington have looked to Wall Street and the ratings agencies to come in and knock heads, but so far it’s the bureaucrats at the Congressional Budget Office—a more successfully nonpartisan analog to the United Nations missions in the Middle East—who have had the most immediate practical impact.

If Obama and Boehner can make a debt deal before next week, and head off the disaster that default would bring, it won’t be because the Republicans, and specifically the Tea Partiers, decided to compromise their position. It will be because Obama and Boehner either found the courage to stand up to the extremists or found a way to defuse their power. If not—well, we all know what it feels like to lament lost opportunities once it’s too late.