The discrepancy between abundances computed using optical recombination lines and collisionally excited lines is a major unresolved problem in nebular astrophysics. Here, we show that the largest abundance discrepancies are reached in planetary nebulae with close binary central stars. We illustrate this using deep spectroscopy of three nebulae with a post common-envelope (CE) binary star. Abell 46 and Ou 5 have O2+/H+ abundance discrepancy factors larger than 50, and as high as 300 in the inner regions of Abell 46. Abell 63 has a smaller discrepancy factor around 10, which is still above the typical values in ionized nebulae. Our spectroscopic analysis supports previous conclusions that, in addition to “standard” hot ({{T}e} ∼ 104 K) gas, there exists a colder ({{T}e} ∼ 103 K), ionized component that is highly enriched in heavy elements. These nebulae have low ionized masses, between 10‑3 and 10‑1 M⊙ depending on the adopted electron densities and temperatures. Since the much more massive red giant envelope is expected to be entirely ejected in the CE phase, the currently observed nebulae would be produced much later, during post-CE mass loss episodes when the envelope has already dispersed. These observations add constraints to the abundance discrepancy problem. We revise possible explanations. Some explanations are naturally linked to binarity such as, for instance, high-metallicity nova ejecta, but it is difficult at this stage to depict an evolutionary scenario consistent with all of the observed properties. We also introduce the hypothesis that these nebulae are the result of tidal destruction, accretion, and ejection of Jupiter-like planets.