Was Bruce Lee actually a good fighter? The question sounds insane, because no one in the history of martial-arts cinema has ever been half as mesmerizing to watch. Plenty of martial-arts stars have speed, but Lee wasn’t just faster than any of then; he had the demonic charisma of speed, a ferocity that charged every jagged movement with expression. His limbs were jackknives on lightning, and his quivering, coal-eyed glower told you how committed he was to every cut and thrust. At the same time, right in the middle of a scene, a part of him hung back and observed it all. That’s why he was the rock star of kung fu, at once in the moment and soaring above it.

But, of course, every time we saw Bruce Lee fighting, he wasn’t really fighting; he was acting. How were his skills in genuine hand-to-hand bloodsport combat? The question is raised — even if it’s not truly answered — in “Birth of the Dragon,” a Bruce Lee biopic set in 1964, two years before “The Green Hornet,” when Lee was an expatriate martial-arts instructor in San Francisco already trying to market himself as a star. (He was born in San Francisco but raised in Hong Kong.) That year, he had a duel — not a fight for show but a real knockdown, no-holds-barred knuckle-bender. His foe was Wong Jack Man, a Chinese master who showed up to challenge Lee. Lee, at that point, was a strict practitioner of the Wing Chun school, but after the bout in question he began to change his method and philosophy of fighting. According to the opening title of “Birth of the Dragon,” this single clash would alter the entire history of martial arts.

That sounds momentous, yet “Birth of the Dragon” is a strange film: It huffs and puffs about what a mythic fight this was, yet it bumbles and stumbles when it comes to showing us what happened, and why it mattered. The director, George Nolfi (a co-screenwriter of “The Bourne Ultimatum” and “Ocean’s Twelve”), treats Lee as a coolly stylized character, a kind of laidback Elvis-in-Chinatown version of the invincible warrior he played on TV and in the movies. But, of course, Bruce Lee wasn’t that character — he was actually a real person! — and “Birth of the Dragon,” which merges a fudged version of the facts with lazy whirling iconography, is neither a compelling biographical drama nor an exciting martial-arts bash. It slips right between the cracks of what a good Bruce Lee biopic should be.

You might assume that the film’s central character would be, you know, Bruce Lee. But you’d be wrong. It’s Steve McKee (Billy Magnussen), a rube from Indiana who travels to San Francisco, tries the Beat scene and the free-love hippie scene (really? In 1964?), and then winds up passing out, drunk, in front of the studio where Lee presides over his martial-arts classes. Lee takes him in and signs him up, and the two become friendly. It’s Steve who makes himself into the liaison between Lee and Wong Jack Man (Xia Yu), a disgraced monk from a Shaolin monastery who shows up in a fedora and fuddy-duddy suit and lands a job in Chinatown washing dishes, which is supposed to be his Buddhist penance for a mysterious sin that he doesn’t reveal until much later.

What Wong is quite open about, on the other hand, is his hostility to the fact that Lee’s students include Caucasians. And that’s a potentially fascinating conflict. When Lee became a superstar, he kicked off an international kung fu craze that had a seismic impact. From that moment on, the Chinese no longer owned this discipline, and the notion that they might have been possessive about it is understandable. But “Birth of the Dragon,” having introduced the issue, barely scratches the surface of it. It’s just a signifier of an idea, a way to set up Lee and Wong as adversaries.

Philip Ng, the Hong Kong-born American actor who plays Lee, has the right face, the right haircut, the right physique — and he’s got a puckish gleam of confidence that’s winning in the way that Lee’s was. Yet unlike “Dragon: The Bruce Lee Story,” the compelling 1993 Lee biopic, “Birth of the Dragon” keeps hitting you with the Lee “mystique” — which is to say, there’s something benignly patronizing about its refusal to allow Lee to become a three-dimensional character. You wouldn’t even know that he was married, and though we see him shooting a low-budget film, and he references the possibility of starring in “The Green Hornet,” what he did to get his showbiz career off the ground remains vague.