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One of those moves was a six-week delay of the deadline, until March 31, for Americans to prove that they have signed up for coverage in time to avoid the penalty for going at least three months without insurance.

The original provisions also included exemptions for illegal immigrants, prisoners, those with religious conscience objections, and people who pay for one another’s medical costs through health care “sharing ministries.”

Supreme Court Justice Sonia Sotomayor on Tuesday granted another exemption for an organization that objected to the contraception mandate tied to Obamacare, saying a group of Colorado nuns who sued over the rule on religious grounds should not have to comply with it until the high court considers the case on its merits this term.

The administrative rule requires large employers to insure birth control as part of their health care plans, including morning-after pills that some religious groups equate with abortion. Republican critics of the law insist that both the contraception and the individual mandate be eliminated. Their own plans for health care reform incorporate some of the Obamacare concepts, but not the mandates. Even a few Democrats are questioning the rule, particularly given the troubles many applicants have experienced.

Sen. Joe Manchin III, West Virginia Democrat, has pushed for a one-year delay of the mandate, saying 2014 should be a “transitional year” for Mr. Obama’s signature reform.

On Monday, former Democratic National Committee Chairman Howard Dean said the mandate “doesn’t really matter that much” but could be politically damaging to Democrats in the midterm elections.

The White House brought some of these doubts on itself in recent weeks by sapping the strength of the mandates.

Under political pressure, the administration said people who lost bare-bones plans because of the law’s coverage requirements could apply for hardship exemptions and acquire basic plans that cover only catastrophic illnesses. The move undercut the administration’s own premise — that low-level plans on the individual market were junk and that the insurance exchanges would provide better, more affordable coverage.

Mr. Mendelson said the move should have a “very small effect” because consumers are likely to find the lowest-tier “bronze plans” on the Obamacare exchange that are similar in price yet offer better coverage than catastrophic plans.

Those who think they do not need health insurance have another reason to procrastinate: The fine for violating the mandate doesn’t have much teeth until the second year.

A single adult who flouts the rule in 2014 must pay $95 or 1 percent of adjusted gross income, whichever is higher. The baseline penalty climbs in 2015 to $325 or 2 percent of adjusted income. In 2016, it will be $695 or 2.5 percent.

The penalties are among a series of taxes and fees designed to pay for the reforms, including government-subsidized plans on the health care exchanges and the state-by-state expansion of Medicaid, which is now optional.

A mandate on large employers was supposed to take effect alongside the individual mandate, but it has been delayed until 2015. Other taxes took effect from 2010 to 2013, and a tax on health insurers takes effect in 2014 — right alongside the individual mandate.

“The mandate is there,” Mr. Mendelson said. “It exists.”