Three judges in London dismissed applications brought by Muktar Said Ibrahim, Yassin Omar, Ramzi Mohammed and Hussain Osman for leave to appeal.
They were jailed for life last July after being convicted at London's Woolwich Crown Court of conspiracy to murder.
Today's ruling was given by Sir Igor Judge, Mr Justice Forbes and Mr Justice Mackay, who heard argument in the case last month.
On 21 July, 2005, the men had tried to detonate rucksacks laden with explosives on three Underground trains at Shepherd's Bush station, Oval station and Warren Street station, together with a bus in Hackney Road, killing themselves and passengers, but the bombs failed to go off.
The attempted attacks came two weeks after four suicide bombers struck in central London, killing 52 people and injuring more than 770.
The judges also dismissed applications brought by Mohammed and Osman against sentence.
Sir Igor said: "These were merciless and extreme crimes. As they were rightly meant to be, the sentences were severe and extreme.
"Beyond doubt, however, they were utterly justified."
At their trial the four had maintained that the events of July 21 were an elaborate hoax designed to protest against and draw attention to Britain's role in the attack upon and occupation of Iraq.
Nigel Sweeney QC, for the Crown, told the judges during the appeal hearing that the conviction applications had "no merit".
Also during the hearing, George Carter-Stephenson QC, for Ibrahim, said to be the leader of the conspiracy, argued that the trial judge "erred in law" in ruling admissible Ibrahim's "safety" interviews - urgent or emergency interviews by police on the grounds of public safety.
The judge also "erred in law in ruling that counsel for the applicant was prohibited from asking questions or calling evidence" about an alleged late confession made by co-defendant Osman.
Grounds argued on behalf of Omar also related to the admissibility of the safety interviews and to the alleged confession.
The judges were told by Michael Wolkind QC, for Omar, that Osman was said to have made admissions to two prison officers.
The "confession" had "amounted to an acceptance by him that the bombs involved were intended to be fully functional".
Omar should have been permitted to challenge that evidence in the normal way.
In the case of Osman, his barrister challenged evidence given by co-defendant Manfo Kwaku Asiedu, which was described as being "central" to his conviction - Asiedu was jailed for 33 years in November after admitting conspiracy to cause explosions.
James Lewis QC had argued it was now clear that "at least in part" that Asiedu committed perjury while giving evidence at the trial.
Complaint was also made on behalf of Mohammed in relation to the trial judge's ruling over the admissibility of his safety interview and the "confession" of Osman.
It was said on his behalf that the only fair course was for the judge to have permitted Mohammed to test the reliability of the confession by questioning the prison officers and, if appropriate, permitting evidence to be called as to Osman's mental state and reliability.
But the judges rejected the grounds of appeal today.
Sir Igor said the jury was given "appropriate and clear directions on all the issues arising in this trial".
Sir Igor said: "On 21st July 2005 London came within a vanishingly short breath of wholesale murder by terrorists.
"The explanation why the date will not be twinned in the annals of its venerable history with the murderous outrage perpetrated on 7th July 2005 is simple: it was sheer good fortune."
He said: "It is axiomatic that every defendant, even a defendant alleged to be involved in direct and dangerous violence on the citizens and institutions of this country, is entitled to a fair trial at which his guilt must be proved.
"This trial was marked with conspicuous fairness and commanding judicial control by Mr Justice Fulford."
Sir Igor said that, after a lengthy examination of the evidence, the court was entitled to record that the defences of the four to the charge of conspiracy to murder "were ludicrous".
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