Charlie Day and Allison Williams in Mystery Dramedy

Speaking to the very cops he’d called to report that someone’s tried to kill him, Jimmy (Charlie Day) suddenly grows panicked. He wants to plead the fifth; he wants to call a lawyer; he’s terrified they’re accusing him of “attempted self-murder.”

The police, understandably, are baffled. The normal term for that is “suicide,” and in any case it’s not anything they’d arrest him for. But Jimmy’s choice of wording is the key to Peter Warren’s directorial debut Kill Me, premiering at SXSW.

To the rest of the world, it may seem obvious that Jimmy cut his own wrists. Jimmy, however, can only understand what’s happened to him as a potential murder, even if it means eventually, reluctantly acknowledging himself as a suspect.

It’s intriguing framing for what could otherwise have been a bleakly earnest drama, conceptualizing depression as an assassin more lethal than any serial killer and one’s own psychology as a mystery more unsolvable than any cold case — with a bracingly morbid sense of humor, to boot. If its exploration of these ideas is ultimately too incomplete to feel fully satisfying, its performances are strong enough to draw attention throughout.

In fairness to everyone who’s not Jimmy, the evidence against him seems crystal clear. The film opens with Jimmy in his bathtub, expelling his last bits of energy to place a 911 call. (That the bathroom looks so dingy he could almost be in a Saw movie is one of many clever production design choices from Ashley Cook.) From his family — which includes his sister, Alice (Aya Cash), his mother (Jessica Harper) and his stepfather (Michael Flynn) — we learn he has a long history of mental illness — notably a very similar incident four years earlier. From the cops, we hear there was no sign of anyone else in his apartment, which, they note, locks from the inside. Not even Jimmy, once he begins frantically scouring his place for forensic evidence, is able to prove otherwise.

Jimmy, however, is resolute that he doesn’t remember doing it. And anyway, why would he? He can’t think of a reason — even as his family points out that he’s been especially down lately, even as he admits to his therapist (Giancarlo Esposito’s Dr. Singer) that he’s stopped taking his meds and even as he’s quick to come up with heartbreakingly mundane justifications for why other people might wish him dead. (Among them: his ex-girlfriend Sarah, played by Sam Rothermel, for not jogging enough, for embarrassing himself at her work party and for not being able to get his dick up that one time.)

Kill Me’s tone veers between dark comedy and even darker drama, and in its goofier moments benefits from Day’s knack for playing guys in the middle of a shrill and wide-eyed freakout. The script, also written by Warren, includes some memorably sharp and funny lines — I laughed at Jimmy, in his initial call to 911, worrying that his blood might stain his bathtub (“Yeah, I think it might,” responds the dispatcher, Allison Williams‘ Margot, after a beat) and his insistence that his dirty apartment is not him living in filth but him living in evidence.

But the role also allows Day to go in sadder, more serious directions, as Jimmy oscillates between his insistent certainty that he’s been targeted and his overwhelming fear that the only person he truly has to fear is himself. Following him through those many mood swings is Margot, a wan, numb soul who has her own reasons for refusing to abandon him. The romantic spark that catches between them is unexpectedly sweet, even hopeful, even as we never lose the uneasy sense that they’re clinging to each other the way shipwreck survivors cling to driftwood.

Kill Me does a fine job of keeping the viewer guessing whether Jimmy’s really onto something (he does eventually come into some clues that could support his theories) or whether he is, as Alice puts it, living “in a Sherlock Holmes fantasy where you just sail on an ocean of delusion.” But while it understands Jimmy’s pain on a bone-deep level, the film occasionally seems almost as cruel to Jimmy as he is to himself.

Over and over, Jimmy is confronted by suggestions that he might not be the only one who suspects he might be better off dead. There’s a suicide victim’s son (David Krumholtz), who spits that “The most selfless act [my dad] ever did was killing himself so my mom and I could move on.” Another suicide victim’s father talks of how his daughter finally “found peace,” of the sort Jimmy complains eludes him. The deep concern Jimmy’s loved ones have for him is clear. The affection they feel for him is less so.

Arguably these are honest if harsh reflections of the way Jimmy sees his place in the real world, and a sharp turn into sentimentality wouldn’t suit a film as prickly as Kill Me anyway. But without any real emotional resolution, the movie ends up feeling incomplete — a bit, perhaps, like a hit job that leaves the victim still gasping for air.