The D.C. Circuit has released a fascinating opinion about the total meltdown of Medicare’s system of internal appeals. The case brings to mind the old paradox of the irresistible force and the immovable object: it’s genuinely vexing and about as difficult to solve.

As I explained eighteen months ago, Medicare’s got a big problem on its hands. A well-intentioned but poorly designed program that enlists private auditors to ferret out waste and fraud has backfired, generating hundreds of thousands of appeals—384,000 in 2013 alone—mainly from hospitals complaining (often rightly) that they’ve been unfairly penalized.

Those appeals have overwhelmed the appeals process in much the same way that hackers can overwhelm a website by inundating it with communication requests. So many appeals have been filed, in fact, that it has forced Medicare to suspend the review process altogether.

That’s the irresistible force. The immovable object is Medicare’s statutory obligation to process appeals within one year. Medicare can’t come close to satisfying that obligation: as things stand, the backlog could take a decade to clear. Medicare’s trying to settle lots of these cases, but so far those efforts haven’t made an appreciable difference.

Frustrated, the American Hospital Association filed a mandamus petition and asked the courts to compel Medicare to process appeals more quickly. The trouble is that CMS is a small, resource-starved agency. Shifting resources into appeals would require CMS to pull resources out of other programs. CMS is understandably reluctant to let other programs go to seed while it hashes out its disputes with hospitals.

What are the courts to do? Sensitive to the agency’s plight, the district court denied the mandamus petition in the hopes that Congress would clean up the mess. But that was more than a year ago. Since then, the problem has gotten worse—and Congress is still sitting on its hands.

In a measured opinion, the D.C. Circuit reversed and told the district court to reconsider whether to grant mandamus relief. After all, the court reasoned, mandamus exists so that courts can compel agencies to discharge their statutory duties. CMS has a clear duty to expeditiously process claims and is failing miserably to discharge that duty.

What’s more, the delays may hurt patients. The court heard from a hospital, for example, that can’t replace ICU beds, modernize a catheterization lab, or replace a roof over its surgery department. And the D.C. Circuit also thought that CMS could (and perhaps should) do more to rein in the private auditors that created the backlog in the first place.

At the same time, the court recognized that mandamus isn’t appropriate for every statutory violation. It’s an extraordinary remedy available only in extraordinary circumstances. “These risks are especially salient here,” the court reasoned, “because mandamus would, in effect, probably require the agency to make major changes to its operations and priorities.”

That risk notwithstanding, the court sent a warning shot across the bow: “[T]he clarity of the statutory duty likely will require issuance of the writ if the political branches have failed to make meaningful progress within a reasonable period of time—say, the close of the next full appropriations cycle.”

In other words, Congress and CMS have been given a brief reprieve. But time is running out to fix the appeals process.