We present new observations of O II recombination lines in ten bright planetary nebulae, along with spatially-resolved measurements of O II and [O III] in the Ring nebula NGC 6720, to study the discrepancy between abundances derived from O II recombination lines and those derived from collisionally-excited [O III]. We see a large range in the difference between O II- and [O III] derived abundances, from no difference up to a factor six difference. The size of this discrepancy is anti-correlated with nebular surface brightness; compact, high-surface-brightness nebulae have the smallest discrepancies. O II levels that are populated mainly by dielectronic recombination give larger abundances than other levels. Finally, our long-slit observation of the Ring nebula shows that the O II emission peaks interior to the bright shell where [O III] and H-beta are strongest. Based on the observed correlations, we propose that the strong recombination line emission in planetary nebulae is a result of enhanced dielectronic recombination in hot gas in the nebular interior, perhaps driven by a hot stellar wind.