Is the aromaticity broken in some resonance structures of para-nitro-aminobenzene?
Your teacher is correct, some of the resonance structures in your figure are not aromatic.  Still though, as resonance structures they contribute to the overall description of the molecule, just less than the structures that are aromatic.  The resonance structures at the right and left end of your figure are aromatic. The fourth resonance structure (the one in the middle) is referred to as being a "quinone" rather than an aromatic.

o- and p-benzoquinone are examples of quinones; they are not aromatic compounds (ref, note the sentence "Quinones are conjugated but not aromatic").



Likewise, their carbon analogues, o- and p-xylylene are also non-aromatic quinoid-like compounds. 

For a compound to be aromatic it must be


cyclic
contain 4n+2 pi electrons
have a continuous loop of p-orbitals
and you must be able to draw double bonds - internal to the ring - from each ring carbon atom.


It is this last point that serves to differentiate quinones and quinone-like molecules from aromatic compounds. 

Finally, aromatic compounds need not be planar, deviations in excess of 15° from planarity can be tolerated and the molecule can remain aromatic (see, for example, [6]-paracyclophane)