  We address the presence of non-distillable (bound) entanglement in natural
many-body systems. In particular, we consider standard harmonic and spin-1/2
chains, at thermal equilibrium and characterized by few interaction parameters.
The existence of bound entanglement is addressed by calculating explicitly the
negativity of entanglement for different partitions. This allows to individuate
a range of temperatures for which no entanglement can be distilled by means of
local operations, despite the system being globally entangled. We discuss how
the appearance of bound entanglement can be linked to entanglement-area laws,
typical of these systems. Various types of interactions and topologies are
explored, showing that the presence of bound entanglement is an intrinsic
feature of these systems. In the harmonic case, we analytically prove that
thermal bound entanglement persists for systems composed by an arbitrary number
of particles. Our results strongly suggest the existence of bound entangled
states in the macroscopic limit also for spin-1/2 systems.
