We compare recent precise/reliable nebular abundances - as derived from high-quality optical spectra and the most recent ICFs - in a sample of Galactic planetary nebulae (PNe) with nucleosynthesis predictions (HeCNOCl) from asymptotic giant branch (AGB) ATON models in the metallicity range Z ⊙/4 < Z < 2Z ⊙. According to the infrared dust features, the sample is divided among carbon-, oxygen-, and double-dust chemistry (CC, OC, and DC, respectively), providing an independent proxy for the nature of the PNe progenitors. Our AGB models, with diffusive overshooting from all the convective borders, nicely reproduce the O overabundances observed in CC PNe, indicating that they evolve from low-Z low-mass (~1 -3 M⊙) AGB stars. This indicates that O is not always a good indicator of the original ISM metallicity and that the O production by low-mass stars should be considered in galactic-evolution models. The lowest metallicity OC PNe evolve from low-mass (~1 M⊙) O-rich AGBs, while the higher metallicity ones (all with uncertain dust classifications) display a chemical pattern similar to the DC PNe. In agreement with the recent literature, the DC PNe mostly descend from high-mass (M > 3.5 M⊙) solar/supersolar metallicity AGBs that experience hot bottom burning (HBB), but other formation channels in low-mass AGBs like extra mixing, stellar rotation, binary interaction, or He pre-enrichment cannot be disregarded until more accurate C/O ratios can be obtained. Two DC PNe show the imprint of advanced CNO processing and deep second dredge-up, suggesting progenitors masses close to the limit to evolve as core collapse supernovae (above 6 M⊙). Their actual C/O ratios, if confirmed, indicate contamination from the third dredge-up, rejecting the hypothesis that the chemical composition of such high-metallicity massive AGBs is modified exclusively by HBB.