The crisis comes days after G4S, the world's biggest security firm, announced it could not provide enough security guards for the London Olympics, forcing the Government to call up 3500 troops to meet the shortfall. Last month MI5 security service head John Evans said the Games offered an "attractive target for our enemies".
Parliament's intelligence and security committee also warned last week that the Olympics had diverted MI5, MI6 and GCHQ from other potential threats, citing the "vulnerability of the UK at this critical period".
The Border Agency official said: "How many other misses have occurred? The missing of counterterrorism 'hits' is a huge thing, but new recruits are not getting enough time to be taught."
"Many of those drafted in to help cope with border staffing shortages are individuals who have been working elsewhere in the Home Office and have received only basic training to work on the airport passport desks. Some have had only a day's training instead of the standard six to eight weeks."
The revelations of lax practice are particularly alarming, coming ahead of pre-Olympic week, starting today, during which the majority of athletes, media and officials will arrive. On July 26 - the day when traffic through Heathrow airport is expected to be most intense - the staff roster shows, according to union sources, that more than half those on duty are relievers.
Counterterrorism police are understood to be urgently seeking a meeting with Border Agency management over the missed alerts.
Chris Hobbs, a retired police officer who spent more than a third of his service working at Heathrow, Gatwick and in Jamaican airports, said: "Missing passengers who were the subject of security alerts was an extremely rare occurrence. The fact that it appears to be becoming a regular event shows that border controls have all but imploded."
Underlining the inexperience of some of the officials now manning borders, a Border Agency document recently distributed to staff reassured new recruits that if they did make a mistake their lack of training would be considered.
Home Secretary Theresa May was under pressure to explain whether she could have averted the Olympic security fiasco as it emerged that the Home Office was warned 10 months ago there were problems with the ability of G4S to provide security for the Games.
A confidential report by the Inspectorate of Constabulary was presented to Home Office ministers last September which should have rung alarm bells about the readiness of the global security giant.
The report by the police watchdog into Olympic security preparedness - which has never been published - raised serious issues over G4S last year, the Independent on Sunday understands. It is not known whether the Home Secretary personally saw the report or whether it only went across the desk of James Brokenshire, her junior minister. But given May's ultimate responsibility for policing and security, she will be under pressure to explain why the crisis came to a head little more than two weeks before the opening ceremony.
The Home Secretary told the Commons last Friday that the shortfall in guards that G4S could provide became clear only the day before, when the Government announced it was deploying 3500 troops to cover the shortage.
May said: "We were receiving reassurances from G4S until very recently, and the absolute gap in numbers was crystallised finally only yesterday."
Opponents point out that her careful choice of words could mean she was aware some time ago there was a general shortfall - raising questions over why it was left so late in the day to act.
G4S chief executive Nick Buckles revealed the firm faces a penalty of up to £20 million ($30.9 million) for failing to deliver on its £284 million contract, as well as paying the Ministry of Defence for providing the troops, meaning a total loss of up to £50 million.
Hundreds of British soldiers face being stuck in Afghanistan for weeks longer than planned because of the knock-on effect of delays to training and holidays.
- Observer, Independent
