It’s possible that the very quality that attracts so many horror directors to tales of demonic possession is exactly what makes so many of them ultimately unsatisfying: Every conflict, and every resolution, can always be written off by resorting to literal deus or diabolus ex machina. Think up a creepy supernatural setpiece with no narrative explanation? “The devil did it” bails you out every time. Write your protagonist into an impossible corner with no reasonable chance of escape? A crucifix or a vial of holy water will do the trick just fine. Relying heavily on this particular crutch, as well as plenty of jump scares and a solid cast, Scott Derrickson’s supernatural horror-policier “Deliver Us From Evil” is a professionally assembled genre mashup that’s too silly to be scary, and a bit too dull to be a midnight-movie guilty pleasure. It should nonetheless cull decent business on Fourth of July weekend.

“Deliver Us From Evil” is being marketed with the enormous above-the-fold tagline: “inspired by the actual accounts of an NYPD sergeant.” The sergeant in question is Ralph Sarchie — also the name of the film’s protagonist — who spent years as both an NYPD officer and an occult investigator, and later wrote a book about his experiences. Still, the description seems to stretch the “based on a true story” credulity limits about as far as they can go. Technically, a film “based on the actual accounts of Lance Armstrong” could omit all mention of steroid use. And a film “inspired by the actual accounts of Lance Armstrong” could theoretically see him partner up with a talking moose to fight crime, just so long as the inspiration was there.

There are no talking animals in “Deliver Us From Evil” — though strange zoological behavior does factor repeatedly into the story — and the proceedings begin on a rather grounded note. Eric Bana stars as Sarchie, a tough-as-nails plainclothes cop with an almost clairvoyant radar for criminal activity, and a wife (Olivia Munn) and daughter (Lulu Wilson) whom he’s increasingly been neglecting in pursuit of Gotham’s worst. He’s partnered with an “adrenaline junkie” officer named Butler (Joel McHale), the type of roughneck who wears a Red Sox hat on duty just to stir up trouble, and prefers to fight perps with a combat knife despite the perfectly good service revolver on his hip.

Patrolling a peculiar microclimate of the South Bronx with nearly nonstop rain and few visible minorities, the two partners investigate a series of rather rough crimes. First, a domestic-violence call leads to a chase after a crazed Iraq War vet (Scott Johnsen). Soon after, the two apprehend a deranged woman (Olivia Horton, all violent twitches and mumbled Doors lyrics) who threw her baby into the lion cage at the Bronx Zoo, leading to a showdown with a few lions themselves. And lastly, an Italian immigrant family reports strange noises coming from their basement, which leads to a gruesome discovery.

The final incident prompts them to investigate a sinister connection between three former soldiers in the Middle East, who were previously seen investigating a mysterious cave in the pic’s prelude. While the film’s opening run falls squarely into police procedural territory, Sarchie begins to notice odd goings-on (strange hallucinations and noises that only he can detect, mysterious Latin inscriptions, barroom jukeboxes still playing the Doors in 2014), and before long he runs across hard-drinking, chain-smoking, bedroom-eyed Catholic priest Joe Mendoza (Edgar Ramirez), who dresses in street clothes and disappoints the eager women at the bars he frequents.

From here the film starts running through rather standard demonic horror paces with one foot still in the precinct, as Mendoza very gradually starts to convince the lapsed-Catholic Sarchie that these cases can’t be solved through criminology alone. But whatever traces of distinctiveness the film had previously shown start to melt away as it trudges toward its finale. It’s hardly a spoiler to note that “Deliver Us From Evil” leads up to a long exorcism scene, though it’s disappointing to see it come and go without adding anything novel to the long tradition of exorcism scenes.

Derrickson, who helmed the well-regarded “Sinister” and “The Exorcism of Emily Rose,” is a thoroughly accomplished orchestrator of jolts and goosebumps, utilizing quasi-found-footage techniques, p.o.v. shots and disquieting overhead angles to good effect. What he’s unable to do here, however, is string these jump scares together to create any sense of overwhelming dread. It’s perhaps appropriate that the film’s most legitimately frightening individual scene, involving Sarchie’s daughter and her collection of creepy toys, seems to come from a different film entirely.

While Derrickson is astute with the horror tropes, the police-procedural elements demonstrate a weird sort of narrative nonchalance. In one scene midway through, for example, Sarchie shows up alone at a woman’s apartment to investigate some strange video files on her computer. Suddenly, the lights start flickering on and off, and Sarchie is viciously attacked by a demonically possessed man who promptly flings himself through a second-floor window. The next scene shows Sarchie calmly back at the woman’s computer. Did the woman notice the commotion in her apartment? Did Sarchie call for backup? Was he hurt? Did he at least help fix the window?

Buried down just below the film’s surface are a number of interesting ideas, such as a conversation between Sarchie and Mendoza contrasting primary and secondary evil, and the implicit conflation of demonic possession with post-traumatic stress disorder, but they’re rarely given much room to breathe. As for the stars, Bana seems fully committed to the character, though his New York accent varies wildly, and Ramirez walks a fine line between low-key naturalism and sleepiness.