Could you say that two electrons in the ground state of a helium atom experience quantum entanglement? They are both in the same energy level and cannot have the same quantum numbers. If one is spin up, the other must be spin down. So, if one "flipped" spins, the other would have to also flip spins or violate sassy Pauli exclusion principle.
In this context it is convenient to look at entanglement as a resource for quantum information tasks. There are several opinions about usefulness of correlations between identical particles as such a resource, but I think the most orthodox one is expressed in this review: http://arxiv.org/abs/1312.4311. The main point follows already from the title: you can extract entanglement from such correlations, but you can't use it otherwise, which means that there is no entanglement.