The issue is not whether some state or most importantly the groundstate of the system is symmetric or not but whether the Hamiltonian commutes with the symmetry operator (like in this case ##\hat{C} \hat{P}##) or not.
You know this from everyday experience: The matter around us is held together by the strong interaction (binding quarks and gluons together to nucleons and these again to atomic nuclei) and the electromagnetic interaction (forming electrons and atomic nuclei to atoms, molecules, and all kinds of condensed matter). All these interactions obey rotation invariance, but the matter around us doesn't only form spheres but all kinds of objects which are not invariant under rotation. So though the relevant Hamiltonian is rotation invariant the states of the matter around us is not.
Within the Standard Model only the weak interaction violates ##\hat{P}##, ##\hat{T}##, ##\hat{C}##, and also ##\hat{C} \hat{P}## (while ##\hat{C} \hat{P} \hat{T}## must be a symmetry in any local relativistic QFT, and there's no hint of violation of this "grand reflection symmetry" at all).
Now take the neutron, which has a spin and magnetic moment, which is an axial vector. If it has also in addition an electric dipole moment, ##\hat{C} \hat{P}## is broken by the dynamics of the system, because there's no other intrinsic vector than the spin of the neutron. If you take into account only the strong interaction which binds quarks and gluons together to a neutron, this implies that the neutron cannot have an electric dipole moment since the strong interaction is invariant under ##\hat{C} \hat{P}## (which is in a way a problem, because there's no "natural" mechanism to explain this invariance of QCD, the socalled "strong CP problem"). Now there's also the weak interaction, which violates CP symmetry due to a corresponding phase in the quark-mixing matrix. You can predict which electric dipole moment you expect from this symmetry breaking, and it turns out to be very small, some orders of magnitude smaller than the upper bound we can get from measurements of the neutron's EDM.
There's much effort to get more and more accurate bounds of this EDM, because we know that CP symmetry should be much stronger broken than predicted by the Standard Model, because CP symmetry is also needed to explain the matter-antimatter imbalance in the universe, and thus it's quite promising to find "physics beyond the Standard Model" looking for stronger CP violation than predicted by the Standard Model.
