Can all quantum superpositions be realized experimentally? When textbooks in QM give example of finite dimensional Hilbert spaces they give examples of photon polarizations or of 2-states systems and sometimes they mention how one can achieve superposition in such cases experimentally. 
On the other hand when they talk about simple potentials like particle in an infinite potential well, and talk about superposition of the stationary states of this problem they never mention how such superposition can be achieved experimentally (despite the fact that we are in the era of nanotechnology and scientists can "engineer" effectively such potentials and many others). Typical example that may appear in textbooks can be as simple as $\Psi(x)=\frac{4}{5}\phi_1(x)+\frac{3}{5}\phi_5(x)$, where {$\phi_n(x)$} are the normalized eigenfunctions of the particle in a box Hamiltonian. Other more general superpositions could be between infinite numbers of stationary states via $\Psi=\sum a_n\phi_n(x)$.
That makes me wonder, is there a fundamental reason that prevents us from engineering such superposition in case of particle in a box and the like, or that we just do not know how to do it yet? why is it possible with spin and seem to be hard with particle in a box? or is it something related to the energy eignestates in the position representation?   
It is kind of frustrating to study for long hours/read/solve problems/HW on all kinds of potentials and on superpositions without knowing how/if they can be realized in experiment.
If someone knows references in which this issue is discussed it would be greatly appreciated.  
 A: It all depends on exactly what you want to do - what system you're handling, what state you want to engineer, and what you plan to do with it. (Note that "I just want to make it" is definitely a perfectly legitimate purpose, but then you also have to think about how you're going to detect it and make sure you've got it!)
For the specific example you pose, creating a superposition of two particle-in-a-box states, you first have to make the box. This is now doable using quantum dots (semiconductor islands in a different semiconductor, possibly with an electron-donating impurity inside) with the right geometry. You also need to make sure that your well is deep enough to accommodate the states you want without shifting their energies too much. After that, though, it's a piece of cake (relatively), since the $\propto n^2$ dependence of the energy levels makes all the transition frequencies distinct. Then you just have to shine a laser pulse at the right frequency and you effectively eliminate all the other levels to get a two-level system interacting with a laser field - a Rabi problem - and you just need to drive a Rabi cycle long enough to get the superposition you need.
However, not all systems are as easily manipulated, and the creation of specific states can be quite challenging. For example, for a harmonic oscillator, all the transition frequencies are the same, and you cannot do this kind of trick, so that making states with a well-defined number of photons/quanta can be very difficult (but doable!). For example, creating superpositions of different coherent states (i.e. "cat states") in light is currently only possible in certain geometries, as I found out on this question. Number states, coherent states, squeezed states, superpositions, entangled states, and so on, have been realized to some degree or other in light beams, mechanical oscillators, atoms and ions, circuit QED, and so on. Again, it depends on what you want your "weird quantum state" to do.
A word of warning, though, on your more general infinite superposition $\Psi=\sum_n a_n \phi_n$. While in principle this is (more-or-less) doable, depending on the state, you also have to bear in mind that one can only ever do a finite number of measurements on the state and therefore you can only ever confirm with certainty a finite number of the coefficients $a_n$. This is another way of saying that you can only ever do stuff with some finite precision. Thus all you can create is a finite sum like $$\Psi=\sum_n^N a_n \phi_n+\textrm{ some amount of noise.}$$
Other than that, it again depends on what system you have and what state you want and it's up to your experimental ingenuity do design a procedure that will take you there.
A: Comment to Shor (apologies for the answer, I cannot yet write comments):
Maybe you are referring to Quantum controllability theorems.
Basically quantum controllability tells you what are the requirements needed for any state of the system to be accessible from any other state by means of an external electromagnetic field at a finite time. The problems are of course related to degeneracies in the spectra of many Hamiltonians. The first papers addressing this problem are J. Math. Phys. 24, 2608, (1983) and Phys. Rev. A, 51, 960 (1995). There are many works after this, particularly due to its importance in Quantum Control and its connection with Quantum Computation.
To Emilio Pisanty: By the way, the harmonic oscillator is a well known uncontrollable system. However any truncation of the Hamiltonian makes the system controllable again.
A: Yes, for any finite-dimensional Hilbert space that effectively describes some physical system, it is possible to design a procedure that prepares the physical system in any state $|\psi\rangle \in {\mathcal H}$.
The precise description of the procedure or gadget is an awkward task because the wording inevitably depends on the physical interpretation of the degrees of freedom and their interactions with various macroscopic fields we have.
However, let me just pick a simple example. $\phi_1$ and $\phi_5$ may be interpreted as the states $|up\rangle$ and $|down\rangle$. Then any linear combination of them, including your combination $0.8 \phi_1+ 0.6 \phi_5$ (incidentally, I also love to use this Pythagorean combination as an example), may be prepared as the state "electron up" with respect to a particular axis that is calculable. I could do the calculation of the axis for you if you want, it's trivial. Of course, the state's phase will be undetermined; the state vector's overall phase is always unphysical (unless you can compare it with another phase of the same system, like in Berry's phase or the Aharonov-Bohm effect etc.).
Quantum computation is a systematic "industry" that is able to perform many elementary operations on the Hilbert space, like exchanging $\phi_2$ and $\phi_5$ if you found it natural. Typically, only several operations – like rotations by preferred angles – are allowed operations on a quantum computer. However, it's straightforward to extend the basic operations of a quantum computer so that you may compose them to any unitary matrix you want. That's also enough to prepare any complex linear superposition of any basis vectors.
It's also possible to "remap" qubits encoded e.g. in many electrons' spins to the amplitudes for any other states even though the detailed "diagram" of the apparatus that achieves such a goal will tend to depend on the precise technical implementation. But in principle, such a "remapping" is analogous to copying the classical information (in bits) from a CD to a Flash memory. The same things may be done at the quantum level but the bits are not copied; the original has to be destroyed.
A: Theoretically, any superposition can be experimentally realized. Experimentally, most can't. The fundamental reason is that a system must be decoupled from its environment so as not to decohere, yet still coupled strongly to an extremely well-calibrated apparatus to generate the superposition. I would guess the 'advanced information' for the 2012 Nobel Prize would be a good starting point, since this issue was so central for both Haroche and Wineland.
As a very rough experimental summary, superpositions of two-level systems has been seen in a huge number of systems, from charge or flux states in qubits to electronic states of semiconductor impurities. A general superpositions on the order of up to maybe half a dozen states has been realized in a few systems. The systems that leap to mind are nuclear spins in a molecule, internal spins in an atom, the spin/electronic state of a chain of ions, momentum state of a free-falling aotm, or the polarization and spatial mode of one (or several) photons.
For a large number of particles, very particular superpositions can be made. For example, spin squeezing has been observed for probably hunderds to thousands of atoms, but a general superposition state of so many atoms is beyond current experiments. That, in a word, is why it's hard to build a quantum computer.
To answer the broader question, superpositions are very useful to know about and understand, even if they're very challenging to make.
