Experimental evidence of Pauli's exclusion principle A fermion is described by a set of quantum numbers, this set of numbers lead us to a unique wave function. If two fermions are described by the same wave function (violating the Pauli's exclusion principle), how can we differentiate this fermions if experimentally it produces the same result? 
 A: I think the atomic configuration is the strongest evidence to Pauli's Exclusion Principle. You simply have atoms that have energy levels sorted in a way that agrees with the pauli's principle. 
Besides, the Zeeman effect shows the separation between different spins, which raises the degeneracy between the levels that agree in all quantum numbers but not the spin.
A: Actually, there is experimental work in progress to test the Pauli principle with higher accuracy ( http://iopscience.iop.org/1742-6596/447/1/012070/pdf/1742-6596_447_1_012070.pdf ): "The experiment VIP at the Gran Sasso underground laboratory is searching for possible small violations of the Pauli Exclusion Principle for electrons leading to forbidden X-ray transitions in copper atoms. VIP is aiming at a test of the Pauli Exclusion Principle for electrons with high accuracy, down to the level of $10^{-29}$-$10^{-30}$, thus improving the previous limit by 3-4 orders of magnitude."
A: Leaving aside the explanation of the valence structure of the atoms, the Pauli principle also explains some of the properties of the nuclei.
Moreover, exclusion leads to the prediction of degenerate gasses, which can be observed directly in a condense-matter context.
A: 
How can we know from the experiment that the Pauli's exclusion
  principle is ok?

This question seems to put the cart before the horse.  The Pauli exclusion principle was formulated to help explain certain experimental results.
From Wiki:

Pauli looked for an explanation for these numbers, which were at first
  only empirical. At the same time he was trying to explain experimental
  results in the Zeeman effect in atomic spectroscopy and in
  ferromagnetism.
...
The Pauli exclusion principle helps explain a wide variety of physical
  phenomena. One particularly important consequence of the principle is
  the elaborate electron shell structure of atoms and the way atoms
  share electrons, explaining the variety of chemical elements and their
  chemical combinations. 

From Encyclopedia Britannica:

Pauli exclusion principle,  assertion that no two electrons in an atom
  can be at the same time in the same state or configuration, proposed
  (1925) by the Austrian physicist Wolfgang Pauli to account for the
  observed patterns of light emission from atoms. The exclusion
  principle subsequently has been generalized to include a whole class
  of particles of which the electron is only one member.

etc.
