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.)
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:
In short: No. For a good coupling coefficient, it even takes a lot of consideration to arrange the windings on one bobbin.
For power inductors (isolated flyback/buck), don't even try. For measurement or signal applications, it might work, but will be very hard.
Edit; two anecdotes:
Since I first wrote this answer, I spent some hours on fixing and recalibrating two old HP 5245L frequency counters (pdf manual, good but large!). These have an internal, oven-controlled crystal reference running at 1 MHz. For the fastest setting of the internal time base, a 2\$\cdot\$5=10 frequency multiplier creates a 10 MHz signal form the 1 MHz signal. It uses three stages, a x2 multiplier, a x5 multiplier and an amplifier. The signal from each stage is coupled out using a transformer built out of two individual and trimmable inductors. Here's a picture of the board, called "A27 MULTIPLIER ASSEMBLY; 1-10MC":
When I started working on this board and figured out how stuff works using the manual, I had to look twice when I read the words "L1-L2; L3-L4; L5-L6 ARE MUTUALLY COUPLED." Now I know it's no fun to read ALL CAPS, but I really like to leave the quote in its original style. Yes, here are three transformers, built from two simple inductors each, accomplishing a good task in a high quality instrument. And the circuit, copyrighted by HP in 1964, still works like new. The manual can be found online in at least two different versions (again: pdf, large!), so I take the freedom to cut-and-paste the part with the frequency multiplier:
Anecdote number two: At a company I worked at, we designed a small auxiliary step-down converter inside of a bigger, industrial power supply. We used two cheap, simple axial inductors; series connected, hoping they would share the large input voltage and survive. Now the circuit, as laid out on the board, with the two inductors aligned in parallel right next to each other, behaved quite a bit differently compared to the first breadboarded version of the design. The reason? With the context above, you are already guessing right, of course: The two inductors, being connected in series but laid out in parallel, had a higher inductance because the two cores were mutually increasing their inductance. This question is on the same topic as this second story.
After these stories, the conclusion is that I still believe my original answer is right. Good coupling coefficient, for power designs, means something (way?!) better than 99 %, and one will need to use a single core and design the windings carefully. For signal applications, you will be able to find examples, and these will be fun to explore.
Most off-the-shelf inductors are "shielded" specifically to prevent one inductor from coupling with another inductor (or anything else).
If you use "unshielded" inductors, or wind your own coils without shielding, or break the shielding off some inductors, all too often you do get unwanted coupling between inductors -- enough to be annoying, but usually not enough to do anything useful.
To get good coupling between coils, you pretty much have to wind them both on the same bobbin -- producing a transformer.
Some examples of coupled inductors made from 2 coils that are different from standard wire-wound transformers (and therefore have worse coupling than a standard transformer):