Earlier this year, the FBI announced that the case had been "solved", but none of the missing works has been recovered. If you visit the Dutch Room of the Gardner today, you will find the frame that once contained Rembrandt's Storm on the Sea of Galilee hanging alongside a portrait by Rubens.
Art crime has long been an obsession for the media, ever since newspaper proprietors discovered that in-depth coverage of the theft of the Mona Lisa from the Louvre in 1911 would dramatically boost sales. Today things are no different: witness the excitement generated by the recent reports about hundreds of works of art looted by the Nazis and hoarded by an octogenarian recluse in Munich.
Or the attention paid to two limited-edition prints by Damien Hirst together worth 33,000 ($66,000) that were stolen from a gallery in London last week.
Invariably, people want to know: what happens to masterpieces after they get pinched? Do they end up in some villain's lair, where a billionaire master-criminal can gloat over them away from prying eyes?
Well, as I discovered while filming the documentary The World's Most Expensive Stolen Paintings, the truth is that they don't.
The reason I wanted to make this programme was to test the validity of the Hollywood myth of art crime, built up over decades in movies from Topkapi (1964) to The Thomas Crown Affair (1999) to Danny Boyle's recent heist flick, Trance (2013). And it turns out that the popular vision of art theft as some kind of glamorous caper is hokum - seductive, yes, but pot-boiler baloney all the same.
The likelihood of a nefarious, Dr No-style aesthete commissioning bespoke, ingenious thefts is slim-to-none: indeed, the very concept of such a figure comes from a tongue-in-cheek moment in the film of Dr No (1962), when Sean Connery visits his adversary's lair on the Caribbean island of Crab Key and spots an easel supporting Goya's portrait of the Duke of Wellington, which had been stolen from the National Gallery in London the year before the movie's release.
The reality of art crime is very different. In the case of the Goya, a 61-year-old unemployed truck driver from Newcastle removed the painting through a toilet window in order to protest against the cost of television licences for pensioners. This is prosaic to the point of absurdity - but most high-profile art crime is dangerous as well as bathetic.
According to Art Theft (2011), by the director of the National Portrait Gallery Sandy Nairne, the international market for stolen art and antiquities is worth as much as $5 billion annually: "This places it among the top international crimes, after drugs, money-laundering and the sale of illegal weapons."
In other words, the notion of a criminal connoisseur in the mould of Dr No may be a myth, but gangs of everyday thieves and thugs do steal art - and lots of it. But since they can't flog stolen masterpieces on the legitimate market, how does the big business of art theft actually work?
There are several ways thieves try to convert their spoils into cash. Newcomers might hope to ransom works back to the institutions from which they were stolen - but this rarely yields results. Often, though, a museum (or an insurance company) will offer a reward for information leading to the recovery of a work. While rewards are never paid to criminals, this can be circumvented with the help of one or two shady middlemen - with the authorities turning a blind eye, as long as the stolen goods are returned.
Most likely, though, pilfered art will accrue value on the black market. Typically, a stolen painting's underworld currency will be between 3 and 10 per cent of its estimated legitimate value, as quoted in the media.
Thus, Vermeer's The Concert, which is often said to be worth up to US$300 million ($366 million), could be a kind of criminal gaming chip, with a felonious value of up to US$30 million. It could then be used as collateral, helping to finance drug deals, gun-running, tobacco trafficking, and other illicit activities.
There are obvious benefits to controlling even a share in a single object worth so much money: "Since the introduction of money-laundering regulations, it has become unsafe for criminals to pay for their operations in cash," says Dick Ellis, who set up the Art and Antiques Squad at New Scotland Yard. "With its black-market value, stolen art can easily be carried across international borders.
"It has an international value, without the hassle of currency conversion, and may even be accepted as a trophy payment by senior cartel members."
Of course, there is also the prospect of a bleaker fate for stolen art: irreparable damage, or, worse, destruction. In 1969, shortly after it appeared in a television programme about Italy's little-known artistic treasures, a late nativity by Caravaggio was stolen from an oratory in Palermo in Sicily.
Given the location of the crime, the most likely culprits were the Mafia. Sure enough, over the years, several Mafia pentiti have spoken out about what happened to the painting, which is still missing.
One said that it was ruined when it was cut out of its frame. Another claimed that it was left to moulder in a farm outhouse, where rats and pigs slowly devoured it, before it was burned.
"The overall recovery rate of stolen art is probably only 15 per cent," explains Julian Radcliffe of the Art Loss Register, an international database of stolen and missing works. "Of the other 85 per cent, probably 20 per cent have been destroyed."
Returning to the Gardner case, if the FBI is so confident that the crime has been solved, then where are the missing paintings?
While filming my new documentary, I met an FBI agent to discuss the possibilities. He told me that he knew who had committed the original crime, and that the agency had even traced the whereabouts of the stolen works until the turn of the millennium, when they had reached Philadelphia. But at that point, the trail went cold.
Does he have any inkling about their location today? "Absolutely not," he said. Nobody outside the criminal fraternity can even say for sure whether they still exist.
Everyone loves a fiendish mystery. But it's time we ditched the Hollywood myths and got real. The truth about stolen paintings is anything but glamorous. Art crime is a brutal business, with repercussions for us all.
