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 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 will 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 $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.