I'm really surprised nobody has mentioned Sybil.
If you're looking for similar movies in the aspect of mental illness then should add these.
Fight Club
A Beautiful Mind
Psycho
Girl, Interrupted
The Three Faces of Eve
Don't Say a Word (in a way, but not as much as the other films)
One Flew Over the Cuckoo's Nest (I haven't seen it but I've been told it's similar)

'Magic", "Tourist Trap" and "May" are definitly similiar but not the "Saw" films. I would also add:
"Dead Silence"
"Love Object"
"The Great Gabbo"

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the first 3 are pretty good suggestions, but I dont understand how saw and saw 2 fit in at all. Donnie Darko either for that matter, but that film doesnt offend me on as many levels as the Saw films, particularly the sequels.
Tourist Trap is fantastic.
Apocalypse Meow

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I haven't seen any of the "Saws", and you would have to pay me A LOT to do such a thing. I can tell you that, to me, they don't look at all similar to PIN.