  Entanglement depth characterizes the minimal number of particles in a system
that are mutually entangled. For symmetric states, we show that there is a
dichotomy for entanglement depth: an $N$-particle symmetric state is either
fully separable, or fully entangled---the entanglement depth is either $1$ or
$N$. This property is even stable under non-symmetric noise. We propose an
experimentally accessible method to detect entanglement depth in atomic
ensembles based on a bound on the particle number population of Dicke states,
and demonstrate that the entanglement depth of some Dicke states, for example
the twin Fock state, is very stable even under a large arbitrary noise. Our
observation can be applied to atomic Bose-Einstein condensates to infer that
these systems can be highly entangled with the entanglement depth that is of
the order of the system size (i.e. several thousands of atoms).
