But since mid-October, HealthCare.gov has been the focus of an aggressive repair effort led by management guru Jeff Zients. On Sunday morning, Zients said fixes and upgrades made by private contractors and government workers had dramatically improved the website, so much so that as many as 800,000 visitors per day would be able to use it as designed.
On Monday and Tuesday, HealthCare.gov handled more than 1.7 million visitors, and officials said the site was working well.
Julie Bataille, a spokeswoman for the federal Centers for Medicare and Medicaid Services, which runs the site, said that 310,000 visitors had logged onto the site in the first 12 hours or so of Wednesday, "which is more than 80 percent higher than a week ago."
Bataille said the site "remains stable" with an error rate of 0.6 percent.
The federal site is selling Obamacare insurance to residents of the 36 states that did not set up their own health insurance exchanges. The remaining 14 states and the District of Columbia have enrolled more than 200,000 people in Obamacare insurance since their Oct. 1 launch.
Despite the fact that the 29,000 enrollments seen Sunday and Monday represents a dramatic improvement over the prior two months, HealthCare.gov and the state-run exchanges remain well short of original enrollment predictions made by CMS.
(Read more: Obama on Obamacare: 'We will make this work')
CMS chief Marilyn Tavenner, in a Sept. 5 memo, noted that projections called for 1.2 million to be enrolled through the Obamacare exchanges by the end of November. Another 2.12 million people were supposed to enroll in the month of December, the deadline for signing up for coverage set to kick in on Jan. 1.
Bataille, the CMS spokeswoman, would not confirm Wednesday that 29,000 enrollments had been made on Sunday and Monday, saying that officials were still "scrubbing" data and would release the next set of enrollment data in mid-December.
Bataille also said that a new "window-shopping" tool on HealthCare.gov, which was put into place over the weekend, is now the site's third-most visited page, with 795,000 users since its launch.
The tool lets consumers views premiums prices, co-payment details, benefit details and provider details of insurance plans offered for sale in their areas of their home states, without having to first fill out an application or confirm their eligibility for government subsidies that could offset the cost of coverage.
CMS had originally strongly resisted suggestions that it implement a window-shopping feature, even when tech experts said that the lack of one was possibly contributing to the massive backlog on HealthCare.gov in its first months of operation. Some state exchanges that allow plan browsing have operated much more smoothly than the federal site.
Bataille also highlighted another new feature of HealthCare.gov, a "reset button" that allows consumers to discard an application if that form has caused the consumer to get "stuck" on the web page, and then start a brand new application.
She said consumers who opt to use paper applications still can use them to apply for coverage that will begin by Jan. 1, saying it takes between three to five days to process such applications. Open enrollment continues through March 31, the date by which nearly all Americans must have some kind of health insurance coverage or face a tax penalty.
The CMS spokeswoman, for the third day in a row during calls with reporters, would not release information about the error rate of so-called "834 forms," the files that HealthCare.gov sends insurers each night with enrollment information from people who signed up for coverage on the site the prior day.
(Read more: SHOP chopped! Obamacare small biz exchange delayed)
Insurers have said that a significant number of the 834 forms they have received to date from the federal site contain errors. Those errors will lead to delays formally enrolling those people in coverage.
CMS on Monday said the tech team had identified a single "bug" that was causing up to 80 percent of the 834 errors. But Bataille, despite repeated requests to reveal what total number the 80 percent statistic was based on, would not give error rate.
"I can appreciate the frustration," Bataille said, when asked Wednesday was the error rate was, and when it was pointed out that her answers to the question were non-responsive.
Bataille also did not directly answer when another reporter asked her if CMS planned, "at some point," to release the error rate.
"I plan to report you on our progress," she said.
Bataille said CMS was in discussions with insurers to monitor the 834 errors, and to make fixes to minimize those errors going forward.
"The vast majority of the fixes needed for the 834 forms are in place," she said. "We are working cooperatively on the issue."
—By CNBC's Dan Mangan and Eamon Javers. Follow them on Twitter