But on Wednesday, hours after that announcement, Sebelius' department sowed further confusion by giving vague answers when reporters asked when the actual deadline for "special exception sign-ups" would be, and when people would no longer be allowed to claim they were eligible for that status.
Officials on Tuesday night said that people who started their application for Obamacare insurance on the federally run exchange HealthCare.gov, which sells plans in 36 states, would be able to enroll past the deadline if they experienced technical problems signing up.
While the site has been running smoothly since December, after an emergency effort to repair its flaws, officials are clearly concerned that a massive influx of consumers before Monday's deadline could leave some of those people unable to enroll by midnight that day.
"We're not going to shut the door on people who were in line," said Julie Bataille, spokeswoman for the Centers for Medicare and Medicaid Services on Wednesday.
The extension is supposed to be limited to only to those people, who theoretically account for a relatively small group of would-be enrollees compared with the 5 million or so people who have already signed up.
But because it is based on the honor system, it could be used by many more people, and possibly help the administration reach its revised goal of signing up 6 million people via state and federal exchanges.
Several states that operate their own Obamacare exchanges and have had ongoing problems with their websites, are granting similar extensions. Cover Oregon, that state's deeply troubled exchange, on Wednesday said consumers there will have until the end of April to enroll.
On Wednesday afternoon, California's Obamacare exchange Covered California followed suit with an extension of its own.
"CoveredCalifornia is not extending its deadline for applying for a health plan," said exchange chief Peter Lee. "However, we're committed to helping all those who started an onlineapplication to cross the finish line."
Covered California in an prepared statement said, "Individuals who start an online application by 11:59 p.m. on March 31, 2014, will have until 11:59 p.m. on April 15, 2014, to complete their application and select a plan.
But others are not extending their deadline, in any way. Access Health CT, Connecticut's exchange, issued a pointed statement announcing its deadline was firm.
The specific rules for the federal exchange's so-called special enrollment period were laid out Wednesday in a posting by the CMS.
Bataille, the CMS spokeswoman, was repeatedly asked during a media conference call Wednesday how, if at all, officials would verify that a person was telling the truth if they said they had tried to sign up by next Monday but were prevented from doing so by tech problems. She suggested there would be no verification process other than consumers "attesting" to that fact.
Bataille noted that the coverage application is an official federal form and "most people are truthful when applying for their benefits."
Bataille also would not give firm dates by which people availing themselves of the deadline would have to attest that they were eligible by virtue of problems with the online exchange, or pick a plan to enroll in.
However, CMS, in a document issued Wednesday, said there was a deadline for people claiming special exceptions who are applying for coverage via paper applications, as opposed to enrolling online. The agency said it will process paper applications received by April 7, and that people whose applications are accepted will have until April 30 to pick a plan.
Brian Haile, who heads tax preparer Jackson Hewitt's health-care reform-related operations, said he was "flummoxed" by news of that hard deadline for paper applicants.
Haile had initially cheered CMS for granting the special deadline extensions because it was originally reported as going until mid-April. Haile has long called for such a deadline because it would give people time to get tax refunds that they could then use to buy insurance.
But by setting April 7 as the deadline for paper applications, it will require people without Internet access or who are uncomfortable with online transactions to mail in their application just a few days after the original March 31 deadline.
Haile said he understood why CMS was being "a little bit coy" in not announcing a hard deadline for online applications under the grace period because officials do not know how many people will end up being stuck in line on midnight next Monday, nor do they know how well HealthCare.gov will handle a possible surge of visitors that strains current capacity.
Kurt DelBene, the senior advisor to Sebelius who is overseeing tech issues for HealthCare.gov, said that as of Wednesday "the site is performing well."
"We've seen only minor issues, all of which have been addressed rapidly," DelBene said.
"We believe the site is prepared to accommodate up to 100,000 concurrent users," he said. "Overall, we feel like we're in very good position to support the consumer demand that we anticipate for March 31."
On Tuesday, visits to the site rose to 1.2 million, according to HealthCare.gov's Twitter account.
Under the Affordable Care Act, most Americans must be enrolled in health coverage by March 31 or pay a penalty equal to 1 percent of their taxable income.
It is not clear what impact the extension will have on getting significantly more people to enroll for coverage this year. A survey released Wednesday by the Kaiser Family Foundation found that 6 in 10 uninsured people are unaware of the March 31 deadline, and half of the uninsured said they had no plans to enroll this year.
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Similar extensions of Obamacare deadlines have been granted in the past. Most notable was at the end of December, when the administration several times effectively lengthened the Jan. 1 deadline, an acknowledgment of the technical problems HealthCare.gov had experienced in its first two months.