Director William Friedkin has confounded fans ever since his now-classic early pictures such as “The French Connection” and “The Exorcist.” Since then, for every middling “To Live and Die in L.A.,” there’s been a tiresome “Rules of Engagement.” After 2003’s underwhelming “The Hunted,” he returns with his most promising effort in years, a story involving paranoia, government intrusion and mental illness.

“Bug” is confounding in its own ways. On the one hand, it’s refreshingly individual. On the other, it seems aimed at people who are not likely to see it, or insulting to those who are, kind of like giving a horticulture seminar at a biker rally. Questions burn at the inflammatory end of this slippery film. Answers seem like conspiracy theories, something this movie revels in.

“Bug” opens without much more sound than the rhythmic pulse of a ceiling fan, mimicked as it is throughout the movie by the whir of helicopter blades as we glide over the silent desert night toward a seedy, isolated motel. The effect is too close to David Lynch for comfort, but Friedkin manages to wring some sweaty anxiety out of it. The inhabitant, Agnes (Ashley Judd), inhales cigarettes and paces the room as the phone rings incessantly. She thinks it’s her no-account ex, who may or may not have been paroled. But whenever she answers the phone, it’s silent.

Perhaps recognizing her loneliness, Agnes’ co-worker R.C. (Lynn Collins) brings her into the orbit of Peter (Michael Shannon). Peter has the look of someone who’s just returned from a lengthy space flight, or at least thinks he has. In reality, he’s just a drifter suffering from his stint in the Gulf War. Peter’s strangeness is a little off-putting at first, but wait until you really get to know him. Shannon, reprising his role from the original version of “Bug,” the off-Broadway hit by Tracy Letts, has a face that looks as if it were carved from a pumpkin, and a personality to match. Just talking about his day is enough to make someone think uncomfortably of the kitchen cutlery.

Peter has nowhere to go. Agnes invites him to stay the night, which leads to more nights, and as Peter grows more comfortable, he’s able to confess a distasteful past. A war veteran, he was wounded and ended up as a lab rat in a military hospital, he says, where they injected him with God knows what. Considering some current headlines, Peter’s disclosure doesn’t sound so wacky. Harder to swallow are his later announcements, like when he reveals that his teeth have been implanted with aphids meant to track his movements. (But ladies, no guy is perfect.)

Popping in and out of this nest of conspiracy is Jerry (Harry Connick Jr.), Agnes’ former husband. In a movie of strong performances, Connick’s stands out. As a bulbous, spiteful lout just out of stir, the former crooner burns up the screen in the small role as a subdued but thoroughly scary sociopath. Jerry, an ignorant criminal, is unburdened by uncertainty and unmoved by Peter’s assertions. “Here I thought you were just a drug addict, and all the while you were looking out for my girl!” Jerry sarcastically exclaims shortly after finding Peter, wrapped in fly-paper strips, peering through a toy microscope for the bugs he claims to have pulled from his epidermis.

Peter may seem ever more nuts, but are we supposed to side with a bullying jerk? Though the answer may appear obvious, “Bug” never completely reveals whether Peter is or isn’t sane. Just when you are convinced either way, an abnormality slips in to shake your faith. Soon Peter is connecting Osama bin Laden to the Unabomber. And yet his doctor, who claims Peter is suffering from paranoia, promises Agnes that the government will return her child, abducted from her years ago, if she’ll help them bring Peter in. What are we supposed to believe?

You can take Peter’s case as an extreme version of a conspiracy theorist, but it’s hard to buy Agnes going off the cliff with him. It’s also insulting to conspiracy theorists. But thinking back on the movie’s points left me with an idea I’d never considered: Conspiracy theories don’t just come from crazy lefties. Aren’t wild, unsupported accusations about WMDs, links between Al Qaida and Saddam Hussein, and Satan’s intent to trap us with sin all conspiracy theories? It’s hard to know just who’s going to stir up a different flavor of paranoia. Undercover aphids don’t really sound so bad, after all. (R) 101 min. S

Ashley Judd (left), Michael Shannon and Lynn Collins illustrate what happens when tickle fights go bad in William Friedkin’s meditation on sanity and conspiracy, “Bug.”