Defence Secretary Sir Michael Fallon has refused to divulge "operational details" of what happened during a Trident test last June.
It is reported that a missile went off course, but the government has not confirmed or denied this.
Theresa May was told about the test when she became prime minister in July, shortly before MPs voted overwhelmingly to renew Trident.
Sir Michael told the Commons he had "absolute confidence" in the system.
Labour and the SNP have urged the government to explain whether the test firing from HMS Vengeance went wrong.
The Sunday Times reported an unarmed missile had been set off from the submarine off the coast of Florida but, rather than head towards Africa, had veered towards the US.
CNN quoted an unnamed US defence official on Monday as saying the missile did deviate from its intended trajectory as part of an automatic self-destruct sequence.
Sir Michael was asked several times by MPs to say whether or not the test missile had gone off course as reported.
He said: "I can assure the House that the capability and effectiveness of of the United Kingdom's independent nuclear deterrent is not in doubt.
"The government has absolute confidence in our deterrent and in the Royal Navy."
For Labour, shadow defence secretary Nia Griffith said: "This is just not good enough."
She added: "At the heart of this issue is a worrying lack of transparency and a prime minister who's chosen to cover up a serious incident, rather than coming clean with the British public. This House, and more importantly the British public, deserve better."
Another Labour MP, Mary Creagh, said a White House official had confirmed to the US broadcaster CNN that the missile did "auto-self-destruct" off the coast of Florida. She asked why people in the UK were "the last to know".
The Defence Select Committee chairman, Conservative MP Julian Lewis, urged the government to be frank about what happened, while the SNP said it was "absolutely outrageous" that information had been deliberately withheld from MPs.
But Sir Michael said: "We do not give operational details of the demonstration and shake-down operation of one of our submarines conducting a test with one of our Trident missiles"
The Ministry of Defence said submarine HMS Vengeance and its crew had "successfully tested" last June, with Sir Michael repeating this.
He cautioned people against "believing everything" they read in newspapers and said: "I am not going to respond to speculation about the test last June."
By Laura Kuenssberg, BBC political editor
It's one of the simplest questions in politics, and one of the most troublesome.
At the start of a critical political week, Theresa May finds herself under pressure for refusing to answer it.
Did she, or did she not know that something had gone wrong with our nuclear weapons, when she asked MPs to vote to renew the costly Trident system?
Scottish First Minister Nicola Sturgeon, a long-standing opponent of Trident, whose submarines are based at Faslane on the River Clyde, called the apparent misfire a "hugely serious issue".
BBC defence correspondent Jonathan Beale said the Royal Navy had carried out half a dozen such tests since 2000 and in the past had publicised successful launches, but this time had not.
Sir Michael told MPs that decisions on publicity were made "on a case-by-case basis" and were "informed by the circumstances".
HMS Vengeance, one of the UK's four Vanguard-class submarines, returned to sea for trials in December 2015 after a £350m refit, which included the installation of new missile launch equipment and upgraded computer systems.
According to the Sunday Times, the unarmed Trident II D5 missile was intended to be fired 5,600 miles (9,012 km) from the coast of Florida to a sea target off the west coast of Africa - but veered towards the US.
In July, days after Mrs May had become prime minister following David Cameron's resignation, MPs backed the £40bn renewal of Trident by 472 votes to 117.
During the debate, Mrs May told MPs it would be "an act of gross irresponsibility" for the UK to abandon its nuclear weapons.
But 52 SNP MPs voted against it, as did 47 Labour MPs including party leader Jeremy Corbyn.
Questioned by the BBC's Andrew Marr on Sunday, Prime Minister Theresa May refused four times to say whether she had known about the test firing ahead of the vote.
Speaking on a visit to Cheshire on Monday, she said: "I'm regularly briefed on national security issues. I was briefed on the successful certification of HMS Vengeance and her crew."
She added: "I have absolute faith in our independent nuclear deterrent. I believe we should continue to have that for the future, the House of Commons voted for that."
What is Trident?
The Trident system was acquired by the Thatcher government in the early 1980s as a replacement for the Polaris missile system, which the UK had possessed since the 1960s.
Trident came into use in the 1990s. There are three parts to it - submarines, missiles and warheads. Although each component has years of use left, they cannot last indefinitely.
The current generation of four submarines would begin to end their working lives some time in the late 2020s.
Earlier, Julian Lewis said Mrs May had been "handed a no-win situation" by her predecessor as Prime Minister, David Cameron, whose "spin doctors" had been responsible for a "cover-up".
He told Today that the government usually released film footage of the "99%" of missile tests deemed a success and that ministers could not "have it both ways" by not announcing when this had not been the case.
But a spokesman for Mr Cameron said: "It is entirely false to suggest that David Cameron's media team covered up or suggested a cover-up for the Trident missile test."
Caroline Lucas, co-leader of the Green Party of England and Wales, called for "full disclosure", adding: "A missile veering off course is deeply concerning. Imagine such a failure occurring in a 'real-world' situation - it could lead to the slaughter of millions of people in an ally's country."
Kate Hudson, general secretary of the Campaign for Nuclear Disarmament, said: "There's absolutely no doubt that this would have impacted on the debate in Parliament."
But former nuclear submarine commander and Ulster Unionist Party assembly member, Steve Aiken, told Today that any fault "would have been sorted out".
"There is a convention that we don't talk about the deterrent... because that is the nature of the deterrent - it is about the security of this nation and I would fully support the prime minister in avoiding those questions," he said.