A key deadline for signing up for Obamacare insurance got a bit squishier Monday, as the nation's biggest insurance marketplace gave some slack to customers enrolling over the phone, and several individual states granted all their customers more time.

Monday is the deadline for people in most of the United States to sign up for individual and family Obamacare insurance plans that take effect Jan. 1.

If people don't meet that deadline, or some of the later ones this week set by individual states, they will have to wait until Feb. 1 at the earliest for their insurance to go into effect.

By midday, the federal insurance exchange, HealthCare.gov, which serves two-thirds of the U.S., appeared to be working smoothly, as did a number of other state-run exchanges.

Nonetheless, within hours, a number of state-run exchanges had effectively extended their Monday deadlines in the face of strong demand, among them California, whose marketplace has the largest number of Obamacare enrollees in the United States

California's exchange will give applicants who began the sign-up process by midnight Monday until midnight Sunday to finish enrollment, according to officials. The extension is being granted on the honor system—the exchange won't be auditing applicants to see if they actually met the criteria.

"We know we've got a lot of people in the pipeline," a spokesman told CNBC.com Monday evening. The spokesman noted that in the past eight days, as the deadline approached, Covered California exchange had signed up 42,000 people, which is more than 80 percent of the number of people who signed up in the first two-and-a-half weeks of enrollment.

"We've just seen a real strong surge," the spokesman said.

Monday remained the deadline for signing up online at HealthCare.gov.

But late in the day, the agency that runs that site announced a grace period for people who were trying to start or finish an application or enroll.

"As we have anticipated, we are seeing very high consumer demand on HealthCare.gov and at the call center for people looking to meet the deadline," said a spokesman for the Centers for Medicare and Medicaid Services.

"Because the call center is experiencing longer-than-normal wait times, some callers are being asked to leave their contact information so they do not have to wait," the spokesman said. "We will call them back at a convenient time starting tomorrow and if they select a plan their coverage will still begin on January 1. Consumers can also shop for coverage and select a plan on HealthCare.gov anytime until midnight Pacific time tonight."

New York's exchange had last week extended its deadline from Monday until next Saturday. The exchange cited snowstorms that slammed the western part of the state recently.

Several other states—Massachusetts, Idaho, Rhode Island, Maryland and Washington—already had extended deadlines as well, to as late as Dec. 23 in some cases.



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Those states got some company as Monday went on.

At Connecticut's state-run Obamacare exchange, "we are seeing very high volume," said Jason Madrak, a spokesman for the exchange earlier Monday,

"As it relates to Web traffic, we typically see between 300-400 concurrent users on our site at any given time on a normal day. Today, we are seeing 750-850 concurrent users," Madrak said in an email. "For call center activity, as of 10 a.m. this morning, we had already seen 4,986 calls come in. To put that in perspective, we had 15,000 calls come in the entire week after Thanksgiving, so we could be poised to handle a week's worth of volume in one day."

By Monday afternoon, Connecticut's exchange announced a "grace period" for applicants.

"If customers have an application started and select a plan before midnight tonight, they can still get coverage beginning Jan. 1, as long as they complete the application and get us all the relevant information by midnight this Friday, Dec. 19," said Jim Wadleigh, acting CEO of the exchange.

Shortly after Connecticut's move, Minnesota's state-run exchange said it was extending its sign-up deadline until 4:30 p.m. Saturday.

We want to give Minnesotans that qualify for financial help as much time as possible to enroll in January 1 coverage," said Scott Leitz, CEO of the MNSure exchange, who added that the extension is not in response to system issues or technical problems.

"TheMNsure system continues to be stable and the vast majority of people coming through the system are doing so without issue," said Leitz. "This change is simply to allow folks that qualify for financial help more time."