They know exactly what the problem is almost immediately, but they can't cure it (but they know exactly where the cure is and who has it). Or they don't and the local shaman/medicine man/crazy Aunt Sadie/homeless man does, and the doctor scoffs at their primitive notions.

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"If this were play'd upon a stage now, I could condemn it as an improbable fiction."
Shakespeare, Twelfth Night
Illuminati's Razor-The most complicatedly evil answer is usually the most correct answer. - Fazor
"Every book is a children's book if the kid can read." - Mitch Hedberg
"Distance doesn’t matter much in space, where if you just start a thing off with the right kind of shove, sooner or later it will get where you want it to go." -Frederik Pohl, Mining the Oort