Experimental limits below predictions
Simple estimates of electron EDMs from Supersymmetry and Minimal Super Symmetric Standard Models are orders of magnitude larger than present experimental limits when superpartner masses are set to 100 GeV and CP violating phases to unity .
Other beyond-Standard-Model calculations that can yield larger than experiment electron EDMs include: left-right symmetric models, lepton flavor changing models, neutral Higgs couplings, dilepton, leptoquark, mirror fermion, horizontal gauge, and composite electron models .
Supersymmetry adapts to experimental limits
Larger superpartner mass and smaller CP-violating phases result in smaller EDMs but also make the theory less attractive. One clever work around is to impose large mass on the superpartners that produce the largest contribution to EDMs while leaving the other particles at low mass .
Another is to adjust the CP-violating phases to cancel out EDMs, but limits from electron, neutron and atom EDMs constrain the adjustments. (See: Can an electron EDM be discovered? )
To explore the vast number of possible couplings in SUSY models, fitting packages have been released ,  where the EDMs of fermions can be generated  or used as input .
Axion-like particles with mass above 0.01 eV, that mediate interactions between electrons and nucleons, which violate parity and time reversal, can give rise to EDMs in atoms and molecules . The effects in alkali atoms are predicted to be large enough that even an improved Cs EDM experiment would be competitive.
Theorests have considered neutron EDMs generated from PeV scale sources of CP violation in PeV scale MSSM extensions . Can electron EDMs be far behind ?
And finally there is always the possibility of a new particle, such as XKCD‘s “The Fixion” that will explain everything .
Why are there no false positive EDM results?
There have been at least 30 electron EDM experiments published since 1962 and more than 50 if one adds neutrons and diatomic atoms. None have reported positive results. Of the eleven competitive electron EDM experiments since 1989, none has a result that is beyond one standard deviation (statistics plus systematics). For the EDM of neutron and the atom, the results are similar.
In the presence of magnetic fields, undetectable by even sensitive magnetometers, and synchronous with the electric field reversal, a false EDM will arise through the magnetic moment. This has always been the challenge of EDM experiments. Can it be that experimenters, as soon as they work on EDM experiments, become so skilled, so clever, so careful, and so prescient, that no false EDM survives?
Symmetry experiments with false negative results
Parity nonconservation (PNC) in atoms, resulting from weak neutral currents, shares many experimental features and a few experimenters with EDM experiments. In 1977 the first two PNC experiments reported null results from measurements of optical rotation in bismuth. The results were published individually and then jointly in the journal Nature. It was not until the third bismuth optical rotation experiment by Barkov and Zolotorev that the effect was observed. There were other false negative results as well.
Additional information and citations are included in the EDM position paper prepared by Gould and Munger for the Nuclear Science Advisory Committee's 2015 long range plan.
PNC experiments differ from EDM experiments in that, even before the first atomic PNC experiment, there was strong high energy physics evidence for weak neutral currents. Presently, there are no high energy physics experiments that are sensitive to EDMs.
Null experiments need confirmation
It is a widely followed practice that important experimental results, positive or negative, be confirmed by another group, preferably at a different laboratory or facility, and preferably using a different experimental method. High Energy and Nuclear Physics have learned this after embarrassing false positive and some false negative experiments. For major experiments, complimentary detectors are built and operated by competing groups.
Interpretation of EDM experiments
The interpretation of results do depend upon the method used to obtain them, especially when the mechanism generating the signal is not known. Listings, such as The Review of Particle Physics , organize results by the method used to obtain them. This is seen in the listing of EDM limits on p 3, where results dating back to 1968 are given. Each particular result represents the most precise achieved using that particular method and/or that particular system.