In letters to Congress and warnings to business leaders about the catastrophic consequences if the United States defaults on its debt, Treasury Secretary Janet L. Yellen has repeatedly offered an important caveat.

She cannot give the exact date when the federal government will run out of cash.

The United States reached its statutory $31.4 trillion debt limit on Jan. 19, forcing the Treasury Department — which borrows huge sums of money to pay nation’s bills — to begin using accounting maneuvers known as extraordinary measures to conserve cash and avoid breaching the cap.

On Monday, Ms. Yellen reiterated previous warnings that the Treasury Department could deplete its cash reserves by June 1. Still, the exact day when the United States will reach the so-called X-date is nearly impossible to determine.

“These estimates are based on currently available data, and federal receipts, outlays and debt could vary from these estimates,” Ms. Yellen has told lawmakers in her letters. “The actual date Treasury exhausts extraordinary measures could be a number of days or weeks later than these estimates.”

While Treasury has the most sophisticated cash management system in the world and employs teams of highly trained economists, its coffers are a blur of payments going out and tax revenues coming in. When its cash balance runs painfully low — as was the case on Wednesday, when the Treasury General Account started the day with less than $100 billion — pinpointing the X-date becomes even harder to predict. In many respects, that is because the moment that a default would occur is a moving target.

Big bills are coming due.

Ms. Yellen has been eyeing early June as a pivotal month since her first warnings to Congress about the debt limit in January. The reason: The federal government spends a lot of money in a short period of time around June 1, and it is impossible to predict exactly how much revenue is going to be coming in and when.

In a report published on Thursday, the Bipartisan Policy Center, a think tank that carefully tracks federal spending, estimated that the government would spend $101 billion on June 1. Most of that money — $47 billion — will go toward Medicare, while the rest will be directed to veterans’ benefits, military pay and retirement, civil service retirement and supplemental security income. On June 2, the government has to pay $25 billion in Social Security benefits and another $2 billion for Medicaid.

During those two days, the government is projected to spend about $140 billion and bring in only $44 billion in tax revenue, leaving the nation’s coffers operating on fumes.

Revenues sputter as refunds flow.

One big problem this year is that tax revenues have been coming in at a more tepid pace than anticipated.

Severe storms, flooding and mudslides in California, Alabama and Georgia this year prompted the Internal Revenue Service to push the April 18 tax-filing deadlines in dozens of counties to October.

Another surprising reason that cash is running lower than some budget experts projected is that the I.R.S. is starting to operate more efficiently. As a result of the $80 billion that the agency received as part of the Inflation Reduction Act last year, it has been able to ramp up hiring and chip away at the backlog of unprocessed tax returns.

Because the I.R.S. has been processing returns more quickly, it is also paying out refunds more quickly and draining the amount of available cash.

June 15 is a critical day.

If Ms. Yellen can find enough coins in Treasury’s couch to pay the bills until June 15, the United States could find itself with a bit of breathing room.

That is because June 15 is when third-quarter payments are due from corporations and people who are required to pay their tax bills throughout the year or choose to make payments every three months to avoid having large bills due in April.

The Congressional Budget Office said in a report last week that an expected influx of quarterly tax receipts on June 15 and the availability of additional extraordinary measures would probably allow the government to continue financing operations through at least the end of July.

The government could receive approximately $80 billion in tax revenue that day. The Bipartisan Policy Center estimates that those funds could be sufficient to keep the federal government afloat until June 30. At that time, Ms. Yellen would also have some additional extraordinary measures at her disposal — a suspension of investments into retirement funds for federal workers — that would allow her to unlock an additional $145 billion and potentially delay a default until well into July.

It’s too close to call.

The lack of clarity about the X-date has made it difficult for lawmakers to know how much pressure they are under to strike a deal. The government may not know how quickly cash is running out until right before the country faces default.

But pressure is still mounting. Congress is likely to take days — if not weeks — to pass legislation to raise the debt ceiling. And even if President Biden and Speaker Kevin McCarthy strike an agreement, there is no guarantee that the House and Senate will easily pass the legislation.

The legislative calendar gets increasingly complicated as summer approaches.

Mr. McCarthy and Senator Chuck Schumer, Democrat of New York and the majority leader, would need to navigate legislation reflecting that agreement through their respective chambers, and the days left to do so are rapidly dwindling. The House is scheduled to be in session for only six days before the end of the month. The Senate is set for just five and is scheduled to be out of Washington beginning on Monday before the Memorial Day weekend.

Mindful that lawmakers are loathe to reschedule their recesses, analysts have been watching the legislative schedule closely as they try to read the debt limit tea leaves. If no deal is signed into law by Memorial Day and Ms. Yellen does not announce that the X-date is delayed, that could raise the likelihood of a short-term suspension of the borrowing cap to give Congress more time to act.

“The congressional calendar is king and will dictate urgency and passage dates for a bill, as has historically been the case,” Henrietta Treyz, the director of economic policy at Veda Partners, said in a note to clients this month.

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