Is it possible to create a coupled inductor (with high coupling coefficient) by just placing two inductors very close to each other on the PCB? (Even as far as placing them on top of each other.)
Answer
Normally you put a magnetic material between them. It becomes a magnetostatics problem instead of an electrostatics.
This core of magnetic material directs all of their magnetic field to be shared, minus a small leakage. You need this core for high coupling coefficient. What you want is called a transformer.
Here is an image of what you need: 
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