Backbone hydrogen bonds between adjacent amino acids in a protein?
The situation you ask about was originally part of the 27 ribbon and 2.27 helix structures considered as possibilities by protein chemists (Linus Pauling?) in the 50s or 60s. The diagram below, showing the ribbon, is from the classic text (long out of print) The Structure and Action of Proteins by Dickerson and Geis.



This turns out to be very rare in proteins. To understand why, one must remind oneself of the conditions for forming hydrogen bonds in proteins. For hydrogen bonds between peptide bonds these would be:


Correct distance between hydrogen of the NH and the O of the CO (too far apart — bond weak, too close — repulsion).
Near(ish) linear angle of the NH and OC.
Lack of steric repulsion of other atoms in the structure


Calculations indicated that a 2.27 helix would be less strained than the 27 ribbon:



Playing with polyglycine in Bosco Ho’s Ramachandran Plot Explorer (Rama) indicates steric repulsion still exists in this structure (white striped connectors):



I could only find one report of this structure, and that was in the artificial peptide Ac-(∆ELeu)5-NHMe, with a model using dihedral angles Φ,Ψ approx. –60°, 30° (as in the Rama simulation) shown below:



It should be noted that this position of the Ramachandran plot is at the margins of stability, and I would expect the bonding only to occur in proteins where it was stabilized by other interactions (as in this artificial peptide) and no lower energy structure were possible.