  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 1.15 days have an abundance discrepancy factor
exceeding 10, and an electron density less than $\sim$1000 cm$^{-3}$; those
with longer period binaries have abundance discrepancy factors less than 10 and
much higher electron densities. We find that [O~{\sc ii}] density diagnostic
lines can be strongly enhanced by recombination excitation, while [S~{\sc ii}]
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,
occurring soon after the common-envelope phase, which ejects material depleted
in hydrogen, and enhanced in CNONe but not in third-row elements.
