Maybe Magic...

[Seiryuu]

I was browsing TVTropes and saw that there were hardly any anime/manga in an interesting trope called "Maybe Magic, Maybe Not". In this setup, you have a series that certainly seems to be supernatural or metaphysical scifi, but we're left guessing for most if not all of the story whether that's actually the case. Anything that absolutely did happen could be coincidence or explainable within real world parameters. Anything that absolutely couldn't have happened without magic, ghosts, gods, aliens, etc. is very shadowed and questionable, so that it could be a mistake or one of the characters' hallucination/delusions. Preferrably the question of whether it's real or not should be a major point. The most significant example in anime/manga, which is already listed at TvTropes, is Umineko no Naku Koro Ni, which after a couple introductory episodes becomes a near-eternal battle of wits between a man who refuses to believe in magic and a witch who needs to prove to him that there's no other explanation in order to survive.

One scenario I've seen a few times in other media that interests me is criminal investigations dealing with impossible, supernatural defense. In these, the start and/or end of the story contains someone who's been arrested for some heinous crime, and for whom all surface evidence points to his guilt, and his insanity based on his absurd claims. However, some third party is investigating, prosecuting or defending the case who starts to see that once you go below the bare surface, evidence doesn't match up at all, and some evidence or subsequent events are unexplainable unless the accused really was on to something. Again though, for most if not all of the story nothing absolutely supernatural happens that can't be explained with science and psychiatry; all that we have is a certain number of impossible clues. Any examples?

[jedinat]

The only thing close that comes to mind is Ghost Hunt. IIRC some of the hauntings seem like they're real, but in the end there's some complicated natural explanation for them. That doesn't last though; lots of magic towards the end.

[Guardian Enzo]

Mythical Detective Loki is fairly close to what you're describing.

[Haak]

Not an anime but...have you seen K-Pax?

I suppose the best one though is Denpa Onna to Seishin Otoko.

[SeijiSensei]

Not an anime, but this movie sounds like exactly what you're looking for. It's available on DVD and Blu-ray from Amazon.

Originally Posted by NY Times

Andy Lau plays Detective Dee, a seventh-century sleuth summoned to the imperial palace to investigate a series of spontaneous combustions. On the eve of the coronation of a controversial empress (Carina Lau), as workers scramble to finish a gigantic Buddha, important officials start bursting into flames. Dee, who had been out of political favor and in prison for several years, sets out to unravel an elaborate conspiracy involving all manner of treachery, hidden agendas, shadowy assassins and an occasional talking animal.

I'm pretty sure that magic doesn't turn out to be explanation for the spontaneous combustion events. I don't what explains the talking animal, though.

[Revelation]

Mouryou no Hako comes to mind, save for various sci-fi elements left as is. A hugely underrated yet surprisingly satisfying anime.

This suggestion may come off as odd, but you can view most of the Black★Rock Shooter OVA as a metaphorical representation of the girls' internal/external conflicts and of what's actually going on, at least that's what I did.

I never really want to suggest Ookami Kakushi, but it does indeed apply pretty well.

Edit: I also want to strongly recommend Gosick, this one fits well, although more in an arc fashion than the entire series.

[SeijiSensei]

I don't know how I could have forgotten Mouryou no Hako. You'll have to look hard to find all the episodes, though. The ones subbed by Aoi have dropped off the regular trackers, but there's a complete version out there on a site I cannot link to.

[Seiryuu]

Originally Posted by Golden Dream

I never really want to suggest Ookami Kakushi, but it does indeed apply pretty well.

Edit: I also want to strongly recommend Gosick, this one fits well, although more in an arc fashion than the entire series.

Why do you not want to suggest OK? That's a good series that I really like, and it most certainly does fit pretty well.
Also, what part of Gosick fits here? Most or all of the "magic" elements seemed to be quickly unraveled and dismissed.

Also, I'm looking for anime/manga, I can find American movies easily enough usually.

Finally, while I referred to the trope "maybe magic, maybe not", I'm including science fiction in the sort of stuff I'm looking for. It doesn't matter to me whether we're dealing with fantasy magic elements or more scifi-type stuff like alien civilizations, wormholes and psychics.

For an example, one semi-movie I saw that made me think of this had a guy find out that his apartment had a gate to a parallel universe, from which a witch was manipulating him and trying to take control of him.

Spoiler:

She ultimately tried to make him kill an infant, but he overcame her control and killed her instead. When he got back to his world though, he found that the witch's familiar had just killed the baby, he got the child's blood all over himself trying to save it, and he was arrested for murder and placed in an asylum. No one believed what he said, but then investigation found evidence that the child really had been killed by an unknown animal, with no participation by him, and his wall was filled with remains of infants dating back decades to centuries. The prosecution had to admit that the only way any of their forensics made any sense was if the guy was telling the truth, but they couldn't do so in court. All they could do was give the evidence to the defense and let them try to convince the jury that witches exist.

Most of the movie deals with the events leading to the man's condemnation, but what really caught my interest was the idea of the investigators gathering evidence and finding that it doesn't add up unless one actually considers that these things might be real.

[Undertaker]

Occult Academy, Paranoia Agent, and Higurashi no Naku Koro ni? might be a good one, otherwise as people already suggested Denpa Onna to Seishin Otoko.

Gosick, would only loosely fit the bill as if you take account of the political stuggle.

Mythical Detective Loki is arguable since the main character is stated as Norse deity Loki as series started despise the way he solves the case involves some unknown element. Though if that series does fit the bill for you then Majin Tantei Nōgami Neuro and Kuroshitsuji (Black Butler) would similar to that.

Also if you haven't seen it, Death Note might loosely fit the bell as well. While it's similar to above three title in that there is supernatural character in it, but the main story is how normal cops connects what's happening to the main character and to the supernatural component of it.