In any sub-Riemannian group one can introduce a Levi-Civita connection adapted to the so-called horizontal subbundle. This can be used to introduce a horizontal connection on a codimension one submanifold and obtain, among other things, a notion of mean curvature. A smooth hypersurface is called H-minimal if its horizontal mean curvature vanishes everywhere. It is natural to pose an analogue of the classical Bernstein problem, starting with the basic prototype of the Heisenberg group. I will discuss a conjecture of Bernstein type in this setting, and show that, despite the fact that H-minimal surfaces are critical points of an appropriate area functional (the H-perimeter), a new phenomenon occurs: there exists smooth critical points of the H-perimeter which are not local minimizers. This is in striking discrepancy with the classical case. I will also discuss in detail how this pathological minimal surfaces suggest an attack to the Bernstein problem, and some interesting open questions.