Or does he? Back in 1991 he was being hailed as the new Sinatra, what with that soft-caramel, jazz voice and that zeal to act tough guys in the movies. Then the next thing anyone knew, Connick was being arrested in the post-Christmas rush at Kennedy Airport for trying to take an unlicensed 9mm gun on board a TWA flight to New Orleans in his spongebag.
Was he trying to out-Sinatra Sinatra? What was going on? Everything went dead after that. Connick simply disappeared. That massive approaching blip on the Big Time radarscope flickered and vanished - for a good 18 months or so.
Flash back five months from the arrest to summer, 1991. Picture Harry at Prince Philip's 70th birthday bash at Windsor Castle, crooning the self-penned "All of You" by special request of the toe-tapping Duke. How many 23-year-olds from the Bayou would have thought they'd be doing the Noel Coward thing for the Windsors in a private party as the sweet Thames ran softly by?
At that particular moment in time, Harry could afford to be a bit smug. He'd recently won a Grammy, had been Oscar-nominated for a song ("Promise Me You'll Remember" from Godfather III), had begun a promising acting debut in Memphis Belle (1990), and had produced a succession of wildly successful jazz LPs (several going multi-platinum, including the sound- track of When Harry Met Sally).
Flash forward. After the airport misdemeanour, the next thing anyone knew about him was a rumour in late 1993 that he had enrolled in a degree course at Georgetown University, to read law. He does now admit that this is so. It seems he lasted only a few weeks of term. What was he thinking? He tells me, unconvincingly, that he had a sudden, inexplicable urge to earn a degree as he "never was educated". Yet his biography shows that from the age of 18 he attended New York's Hunter College and the Manhattan School of Music - where he learnt, among other things, the complex and highly cerebral art of orchestration.
He claims he hates the work of his fellow New Orleans denizen Anne Rice, but I suspect that Harry has a southern Gothic streak to his soul, souped up by an Irish-Jewish gene-pool inheritance.
Try Jewish alienation and Irish melancholy. Perhaps a race memory surfaced as he tickled the ivories on that sunny Berkshire afternoon: the pogroms of Prince Philip's Romanov forbears, Cossacks riding down Connick's distaff ancestors back in old Kiev. Whatever.
"I find dark themes attractive," he tells me, conspiratorially. "I've seen a lot of darkness and I find it more interesting. When I sing "Oh I Love, You!" - that's nice - but when I sing about death, disease, I like that. I'd love to play slow tragic songs all day; I love it".
Sandra Bullock has mentioned that "there's so much pain in Harry's past", referring to the death of his mother of ovarian cancer when he was only 13. It affected him deeply. Harry himself has even speculated that the overtly "boyish" side of his nature is the result of some kind of "arrested development".
I asked Harry whether his mother would have liked Hope Floats, in which he plays a blue-collar Texas stud who saves the returning prom queen Sandra Bullock from the despair of a wrecked marriage (the sort of role a pre- lapsarian Rock Hudson so often performed). Harry doesn't blink as he replies. "Maybe not," he concedes, with unexpected truthfulness. "It borders on sennimentality and she was not a fan of that." Hooo! This man is wonderfully off-message for someone on a slick PR junket ("I think there were a lot of things that were undeveloped about my character," he tells me later, "which was kind of frustrating.")
His mother would have liked his celebrated role as a serial killer in Copycat, he says with some confidence. And no doubt his next role - as a rapist being let out of prison, would also intrigue her, as a former state judge from NYC, "whose favourite colour was black".
I ask him about the gun. He stalls. "You can get my statement from the police department," he says. Your "missing years"? He doesn't acknowledge them. Instead, he laughs, waves his big hands, does his gregarious, love- me Harry thing. Whatever his mysterious crisis was, it ended in 1994, with marriage, a new album and a US tour happening almost simultaneously.
The triple-platinum ambition is untarnished. It's telling when he finishes a question for me. "It seems to me that however fine your music is..." "It's not gonna get any bigger, yeah." Which seems to indicate a certain frustration with the genre of music for which he is the crown prince, big-band swing. It's clear that he's really been mulling it over in his head, and it's been bothering him somewhat. Move into movies, he's thinking, since you don't have it in you to make a pop record, and make it really big-time like your fellow top-gun in Independence Day, Will Smith (a cameo role there for Harry). Would he like to be Will Smith? "Sure, he gets $20m a movie, that's what I'm working so hard to achieve." He visibly collects himself. "Not for the $20m," he adds, hastily, "but for the opportunity to select what I want to do." Then what? The canvas chair and the riding crop. "My ego and my ambition wants to direct," he says. "But my head says wait until you have enough skill."
It's that Harry career tornado again.
`Hope Floats' is released on 13 November