$
\newcommand{\CP}{\mathit{CP}}
\newcommand{\fm}{\text{fm}}
\newcommand{\efm}{\,e\,\fm}
$The current state of the art is that we predict the neutron has a nonzero electric dipole moment, but seventy years of efforts to measure the neutron's EDM have been consistent with zero at higher and higher precision. Dipole moments are measured with units of $\text{charge}\times\text{length}$, so a "natural unit" for the neutron's EDM would involve the fundamental charge $e$ and the neutron's radius, a femtometer. In these units the current upper limit is $d_\text{n} \lesssim 10^{-13}\efm$; that is, any nonzero dipole moment is contributing at the sub-part-per-trillion level of the scale you'd predict from dimensional analysis.
It's worth taking a minute to think about just how small that is. If you wanted to create a dipole moment this size with two hypothetical unit charges in the neutron's core, a convenient visualization is that $10^{-13}$ times the Earth's radius gives the wavelength of red light. This is a visualization used by people who think of the EDM as a property of the shape of the neutron's charge distribution.
Or suppose you wanted to think in terms of Feynman diagrams for vacuum polarization, where an electrically neutral particle spends part of its time as a virtual pair of charged particles. You would probably have to draw at least ten trillion such diagrams for the neutron (including diagrams for the electromagnetic, strong, and weak interactions) before you found one that made a net contribution to the neutron's EDM, because those diagrams involve particles with unit-scale charges and neutron-sized lengths.
In order for any "object" to have a dipole moment, there must some separation between the object's positive and negative charge distributions. How could this be, though, when the neutron is an uncharged particle?
The neutron has zero net charge, but it has a nontrivial charge structure which is best explained using the quark model. The most obvious consequence of the neutron's internal charge structure is its nonzero magnetic dipole moment, whose natural unit $\mu_N = e\hbar / 2m_\text{nucleon}$ correctly sets the scale. Note that the value of the nuclear magneton is $\mu_N \approx \frac{1}{10}\efm \cdot c$, so another way to point out the astonishing smallness of the neutron's EDM is to slip into $c=1$ language and say that $d_\text{n}$ is at least a trillion times smaller than $\mu_\text{n}$.
The reason for the shocking asymmetry between the electric and magnetic sectors is that $\vec d_\text{n}$ and $\vec \mu_\text{n}$ behave differently under $\CP$ transformations. The nucleon is the ground state of quantum chromodynamics, so all of QCD's symmetries correspond to good quantum numbers for the nucleon. The electric monopole and magnetic dipole are $\CP$-even, so nucleons can have nonzero values for those moments. The magnetic monopole and the electric dipole are $\CP$-odd, so only a hypothetical $\CP$-odd particle could have nonzero values for those moments at the "natural" scale. (This related answer begins with an important clarification.)
Why do we have reason to believe a meaningful neutron EDM exists?
The $\CP$ transformation is a mathematical procedure which transforms our model of a matter particle into our model of its antimatter partner. When we say "this theory is symmetric under $\CP$," we mean, explicitly or implicitly, that the theory predicts matter and antimatter will evolve the same way under the same conditions. We therefore have experimental evidence that our actual universe is not symmetric under $\CP$: our universe is full of matter baryons, but antimatter baryons occur only rarely and briefly.
And in the spirit of Gell-Mann's totalitarian principle, if there is $\CP$ violation anywhere in the universe, then there is $\CP$ violation in the neutron. It leaks in by the same mechanism as do corrections to the magnetic dipole moment, a mechanism which was already mentioned above in the context of "vacuum polarization." If we have the same vacuum today that we had during baryogenesis, then whatever $\CP$ violation gave us a matter-filled universe is still happening deep down in the belly of a neutron interacting with an electric field.
And if we can detect that $\CP$ violation in the neutron, it will tell us something about the baryogenesis epoch.
(Here are
some
related
answers
I've written about using the same trick to study the $P$-violating weak interaction.)
We have already discovered some violation of $\CP$ symmetry in the Standard Model. That known $\CP$ violation corresponds to a Standard Model prediction for the neutron's EDM of $d_\text{n}^\text{SM} \sim 10^{-17}\efm$, which is four orders of magnitude below the current limit. However, the current Standard Model also underpredicts the $\CP$-violating baryon asymmetry of the universe, and by a comparable factor. This makes the current generation of nEDM experiments very interesting, whether they finally see a nonzero effect or whether they continue to set more-stringent upper limits.
How is it possible for such a thing to exist?
We don't know.
We know the mechanics. The net-neutral neutron has a complicated electric charge distribution, which can be polarized and which could in principle be polarized permanently. (This is a slightly different argument than for the net-charged proton and electron, whose permanent EDMs are forbidden by the same symmetry arguments but for which an EDM could be modeled as a displacement between the charge distribution and the mass distribution.)
But we don't know the details. The community which measures neutron EDMs like to think of themselves as "theory killers," because there are dozens of interesting ideas about physics which have been abandoned for predicting unphysically large neutron EDMs. Next on the chopping block is supersymmetry; a good fraction of the phase space for supersymmetric $\CP$ violation has already been ruled out by the continued non-observation of a permanent neutron electric dipole moment.
