The discrepancy between abundances computed using optical recombination lines (ORLs) and collisionally excited lines (CELs) is a major unresolved problem in nebular astrophysics. We show here that the largest abundance discrepancies are reached in planetary nebulae with close binary central stars. This is illustrated by deep spectroscopy of three nebulae with a post common-envelope (CE) binary star. Abell 46 and Ou5 have O++/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, but still above the typical values in ionized nebulae. Our spectroscopic analysis supports previous conclusions that, in addition to "standard" hot (Te~10000 K) gas, a colder (Te~1000 K) ionized component that is highly enriched in heavy elements also exists. These nebulae have low ionized masses, between 0.001 and 0.1 solar masses 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, in post-CE mass loss episodes when the envelope has already dispersed. These observations add constraints to the abundance discrepancy problem. Possible explanations are revised. Some 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 the observed properties. The hypothesis that these nebulae are the result of tidal destruction, accretion and ejection of Jupiter-like planets is also introduced. ; Comment: 24 pages, 5 figures, 8 tables. Accepted for publication in The Astrophysical Journal