Can we / should we use the Pauli principle to explain band structure? Checking the wikipedia article on band structure, I got caught in major doubts...
They try to give an intuitive explanation of the band structure relying heavily on Pauli Exclusion Principle:

if a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms' atomic orbitals overlap. Since the Pauli exclusion principle dictates that no two electrons in the solid have the same quantum numbers, each atomic orbital splits into N discrete molecular orbitals, each with a different energy. Since the number of atoms in a macroscopic piece of solid is a very large number (N~1022) the number of orbitals is very large and thus they are very closely spaced in energy (of the order of 10−22 eV). The energy of adjacent levels is so close together that they can be considered as a continuum, an energy band.

I... am confused. For me,  the Pauli exclusion principle states the Following:

2 identical fermions in the same physical system can not be in the same state at the same time

The state of a particle is described by its wave function. Therefore, I don't see how the Pauli principle could Apply to the orbitals of two atoms "far away" in the Crystal...
I mean, to illustrate, if we imagine a 1D monoatomic crystal and take a string of 3 atoms that we label A, B, and C:
$$A-B-C$$
I could more or less imagine that the overlap of the valence orbitals of A and B could imply that the Pauli exclusion comes into play for the valence electrons in the A-B system (although I am already not sure this is the right way to describe this). 
However, why would the Pauli principle say anything about the electrons of A and C? I mean, the wavefunctions of the electrons of A and C do not overlap. Being "distant from each other", the spatial part of $\psi$ is distinct for those electrons, and therefore the states are already different. No need to call on the Pauli principle, which in this case, should provide no information. No?
Why would the Pauli principle cause a never-ending splitting as we add more and more atoms to the crystal whose electrons have... nothing to do with each other as the distance increases? (ignoring the fact that they could eventually be delocalized as conduction electrons in a metallic crystal) 
The way to introduce band structure that I have studied does not make use of the Pauli Principle (in fact, in the book I am thinking of, a chapter on the band structure is placed before the one tackling identical particles). Actually, we could establish the band structure in a single-electron approximation, which suggests that the exclusion principle would have Nothing to do with this result. It only relies on the translational symmetry of the Crystal, deducing the Bloch states and injecting them in the SE to show the bands $E_n(k)$ emerging as a solution.
I understand that the exclusion principle will be important to describe how those bands get filled, but it should not be necessary, in my understanding, to explain that they exist.
I find it highly surprising to find an explanation based on a radically different principle, and I have a hard time imagining that both ideas are equivalent.
Is this Pauli principle approach really correct? If yes, what did I misunderstand?
If yes, can we show that it is equivalent to the well-known periodical-Hamiltonian demonstration?
 A: You are correct and the wiki article is problematic. The discreteness of the energy levels within a band are not due to the Pauli principle but due to crystals being finite in size leading to discrete values of the wave numbers $k$. As you said the Pauli principle just limits how many electrons can fill a band.
See for example: Energy Bands in Crystals 
A: You wrote in your question:

I mean, to illustrate, if we imagine a 1D monoatomic crystal and take a string of 3 atoms that we label A, B, and C:
  $$A-B-C$$
I could more or less imagine that the overlap of the valence orbitals of A and B could imply that the Pauli exclusion comes into play for the valence electrons in the A-B system (although I am already not sure this is the right way to describe this). 
However, why would the Pauli principle say anything about the electrons of A and C? I mean, the wavefunctions of the electrons of A and C do not overlap. Being "distant from each other", the spatial part of $\psi$ is distinct for those electrons, and therefore the states are already different. No need to call on the Pauli principle, which in this case, should provide no information. No?  

I think here lies the core of the problem. The atom wavefunctions do not overlap, but the wavefunctions of all valence electrons form one "overarching" wavefunction. You can't tell anymore where every electron in this chain is to be found. For example, the valence electron initially in $A$ can (when brought together with the other atoms to form a chain) be found in $C$, $D$...etc. The same holds for all other valence electrons. And because they are indistinguishable the Pauli Exclusion Principle causes half of all the valence electrons to be in different energy states and form a band.
A: The excerpt from the Wikipedia article is wrong beyond repair, indeed. My advise is to look for another source of knowledge, preferably a reputable textbook.
Here is, at the request of commenters, not the OP, very simplified a count of band structure. 
What is important is that the bandwidth is caused by the variation in crystal kinetic energy. Atoms have shells due to the Pauli principle. Consider sodium and let us ignore all electrons except 3s. The, simplified, electronic orbitals of a 3D periodic system of sodium atoms, are the linear combinations of the individual atoms. The complex coefficients can be written, up to normalisation, as $e^{i\vec k_j \cdot \vec r} $. When we apply the kinetic energy operator we find that these states have crystal kinetic energy $|k_j|^2 /2m$. The states with the lowest kinetic energy are filled with 2 electrons each. In the case of a semiconductor the bands are formed from bonding and antibonding states, leading to the completely filled valence and the completely empty conduction bands, at 0K.
A: The Pauli principle is indeed very important in explaining how the band structure forms the way it does. Two Fermions cannot have all quantum numbers with the same values. Remember the energy is also a quantum number. The reason why electrons do not collapse to one flat band because electrons obey the Pauli principle. They avoid each other by differing their energy eigenvalues. If the number of electrons is in order of Avogadro number, they only have a little room left in the energy space, and therefore their energy will only slightly differ, and thus thus semi-continuum energy bands.
Also, Pauli principle is also implicitly included in the calculation of Bloch states through the implementation of Slater-determinant!
However, you can certainly calculate a single-particle energy profile $E_n \left(k\right)$ without Pauli principle (recall Pauli principle is only for two or more fermions). However, if you combine it with the Pauli principle you can even use it as an approximation to many-electron band structures. Here, what I mean by "structures" certainly includes the arrangement of electrons in the energy space; i.e band filling, as "structure" is indeed a synonym for "arrangement"! The case will be different for many-boson systems which will have a single flat (not parabolic-like) band, even though the individual particle have a parabolic-like energy profile. Also, here we are talking about ground-state bandstructures!
A: E. Bellec left a commment to your question, where he mentioned the behavior of matter near zero Kelvin. In Einstein-Bose condensates

a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent macroscopically.

In such condensates three things happens:


*

*In dependence from the reached temperature near zero Kelvin a certain part of electrons in the atoms are in the lowest energy state.


*

*The termic energy is removed to a great extend from the condensate and the the disorder from the vibrating subatomic particles is suppressed.


*The magnetic dipoles of the largely immobile atoms get aligned. 


Let’s just focus on the magnetic dipoles. The magnetic dipole moment is an intrinsic property of the involved subatomic particles and is not lost on higher temperatures. The self-alignment of these dipoles in largely immobile atoms gets destroided in a higher temperature surrounding through the emission and absorption of photons. BTW, this is somehow similar to the destruction of permanent magnets under higher temperatures.
With that background, let’s see what Pauli discovered:

that two or more identical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously.

The quantum system is the atom or a molecule and the fermions are the electrons in the shells. Call it the electrons spin or the electrons magnetic dipole moment, they are the reason for how electrons behave like they behave in atoms. Pauli’s exclusion principle states this phenomenon, but not explain it. 
Just to get a better idea, put the spotlight on the electrons magnetic dipole moment (they are correlated one by one with the spin).
These tiny magnets are self-arranged around the nucleus and without distortion from photonic interactions they would form an ideal Bose-Einstein condensate. Even atoms with odd number of these magnets (shortened for “these electrons with their spins aka magnetic dipole moments”) arrange themself in pairs and than behave like bosons. It’s a question of non-disturbness from the surrounding energetic influences.
Does a metal behave like a quantum system? Near zero Kelvin yes, it behave like a system in lockstep.and, to underline it, all the atoms have their electrons in the lowest possible state. There is no question about a band structure. With higher temperature the electrons are less bonded to the nucleus and for some elements or compounds the electrons are not immobile and this degree of freedom is called a band structure. The Pauli principle does not have to do with that anything.
