  We have studied the magnetic properties of Na$_2$Co$_2$TeO$_6$, which
features a honeycomb lattice of magnetic Co$^{2+}$ ions, through macroscopic
characterization and neutron diffraction on a powder sample. We have shown that
this material orders in a zig-zag antiferromagnetic structure. In addition to
allowing a linear magnetoelectric coupling, this magnetic arrangement displays
very peculiar spatial magnetic correlations, larger in the honeycomb planes
than between the planes, which do not evolve with the temperature. We have
investigated this behavior by Monte Carlo calculations using the
$J_1$-$J_2$-$J_3$ model on a honeycomb lattice with a small interplane
interaction. Our model reproduces the experimental neutron structure factor,
although its absence of temperature evolution must be due to additional
ingredients, such as chemical disorder or quantum fluctuations enhanced by the
proximity to a phase boundary.
