It has recently been noted that there seems to be a strong correlation between planetary nebulae with close binary central stars and highly enhanced recombination line abundances. We present new deep spectra of seven objects known to have close binary central stars, and find that the heavy element abundances derived from recombination lines exceed those from collisionally excited lines by factors of 5-95, placing several of these nebulae among the most extreme known abundance discrepancies. This study nearly doubles the number of nebulae known to have a binary central star and an extreme abundance discrepancy. A statistical analysis of all nebulae with measured recombination line abundances reveals no link between central star surface chemistry and nebular abundance discrepancy, but a clear link between binarity and the abundance discrepancy, as well as an anticorrelation between abundance discrepancies and nebular electron densities: all nebulae with a binary central star with a period of less than 1.15d have an abundance discrepancy factor exceeding 10, and an electron density less than ~1000cm^-3^; those with longer period binaries have abundance discrepancy factors less than 10 and much higher electron densities. We find that [OII] density diagnostic lines can be strongly enhanced by recombination excitation, while [SII] lines are not. These findings give weight to the idea that extreme abundance discrepancies are caused by a nova-like eruption from the central star system, occuring soon after the common-envelope phase, which ejects material depleted in hydrogen, and enhanced in CNONe but not in third-row elements.