Once upon a time heist movies were about actual human beings pulling off ingenious capers that were explained to the audience in intelligible terms. Even at their most farfetched, the best of them required an inner logic that made them satisfying diversions. Nowadays, however, special effects are the key, and the human element goes pretty much out the window. That’s certainly the case with “Now You See Me,” a fast-moving puzzler that, on a minute to minute basis, is moderately engaging but by the close has become a rather heavy-handed bit of hokum with lots of plot holes left unexplained.

The realm of big-money Vegas magic shows is the milieu the script employs. Four sleight-of-hand artists—arrogant industry staple J. Daniel Atlas (Jesse Eisenberg), sarcastic mentalist Merritt McKinney (Woody Harrelson), shapely escape artist Henley Reeves (Isla Fisher) and street practitioner Jack Wilder (Dave Franco)—are recruited by a mysterious mastermind to pull off a series of high-profile acts as a team. Before you can say hocus pocus, they’re bankrolled by international insurance mogul Arthur Tressler (Michael Caine), and after the first show results in the theft of a bundle of Euros from a Paris bank, FBI Agent Dylan Rhodes (Mark Ruffalo) is assigned to crack the case, partnered by pretty Interpol representative Alma Dray (Melanie Laurent). For inside information on the tricks of the trade they turn to professional magic debunker Thaddeus Bradley (Morgan Freeman), who’s made a career of revealing how magicians do what they do to the general public—and has ruined some of them in the process.

But Rhodes still finds himself in over his head, unable to prevent further gambits that result in massive unauthorized bank transfers and the theft of a huge safe from a supposedly secure warehouse. Everything comes to a head in New York City, where the final confrontation involves a fight in an apartment building, a car chase that ends in flames on a bridge, an outdoor rooftop performance attended by thousands of viewers, and a twist that not only reveals the identity of the mastermind behind everything but the motivation as well.

On the most basic terms, the mazelike scenario of “Now You See Me” is a pleasantly convoluted contraption that’s fair enough in what it ladles out about the “why” of what’s happening (indeed, it would take a pretty dense viewer not to know well in advance the motive at work here), but keeps the precise “who” in reserve to the last reel, although the old mystery-story staple of always assuming the least likely person to be the perpetrator remains a safe bet here. But the “how” of each step along the way isn’t always covered satisfactorily. To be sure, Bradley offers blow-by-blow explanations for some of the shenanigans (Freeman’s flashback recitations are reminiscent of the ones George Peppard offered in every episode of the old “Banacek” television series), and they’re nicely edited by Robert Leighton and Vincent Tabaillon. But many other feats are left unexplained (papered over by “magic” effects), and when you come to the end—which also involves some sort of secret association of magicians that’s existed for centuries—the overall scheme, based on coincidence after coincidence and a gargantuan amount of implausibility (not least about the mastermind’s identity), seems in retrospect a very incredible invention indeed.

On the plus side, though the characters are basically caricatures, the cast is agreeable, with Harrelson having a field day as the snide mentalist, Eisenberg replaying his “Social Network” persona to good effect, and Fisher providing nifty eye-candy. Franco gets the short end of the stick among the Four Horsemen, as they’re called, but he handles what the script provides adequately enough. Freeman offers his patented brand of calm smugness, but Caine is wasted in a stock role that frankly doesn’t make much sense—why would such a tycoon bankroll a magic act? Ruffalo appears to be radically overdoing things through much of the running-time as the overexcited FBI man, but there’s a method to his apparent madness, and Laurent is likable romantic interest for him.

Director Louis Leterrier helps matters by staging it all at a lightning pace, in obvious hope that pushing rapidly ahead will keep viewers from thinking too closely about unlikely plot developments. Technically the picture is first-rate, with Michael Amundsen and Larry Fong’s widescreen cinematography giving the images a lush, colorful look and Peter Wenham’s production design effectively elaborate, especially in the highly theatrical scenes of the quartet in performance, where the snazzy lighting effects and whiplash camerawork fulfill their purpose. Brian Tyler’s music score is also appropriately flashy.

“Now You See Me” constantly indulges in the misdirection that Bradley repeatedly refers to as the mainstay of the magician’s art, and as Eisenberg’s Atlas is accustomed to say of any trick, if you look at it too closely you’ll miss the forest for the trees. But though the forest, when it’s revealed at the close, doesn’t really amount to much, the trees afford occasional moments of pleasure along the way.