Aims: The chemical behaviour of an ample sample of planetary nebulae (PNe) in NGC 6822 is analysed. Methods: Spectrophotometric data of 11 PNe and two H ii regions were obtained with the OSIRIS spectrograph attached to the Gran Telescopio Canarias. Data for other 13 PNe and three H ii regions were retrieved from the literature. Physical conditions and chemical abundances of O, N, Ne, Ar, and S were derived in a consistent way for 19 PNe and 4 H ii regions. Results: Abundances in the PNe sample are widely distributed showing 12 + log (O/H) from 7.4 to 8.2 and 12 + log (Ar/H) from 4.97 to 5.80. Two groups of PNe can be differentiated: one old with low metallicity (12 + log (O/H) <8.0 and 12 + log (Ar/H) < 5.7) and another younger one with metallicities similar to the values for H ii regions. The old objects are distributed in a larger volume than the young ones. An important fraction of PNe (over 30%) was found to be highly N-rich (Peimbert Type I PNe). Such PNe occur at any metallicity. In addition, about 60% of the sample presents high ionization (He++/He ≥ 0.1), possessing a central star with effective temperature higher than 100 000 K. Possible biases in the sample are discussed. From comparison with stellar evolution models by Karakas (2010) and Fishlock et al. (2014) of the observed N/O abundance ratios, our PNe should have had initial masses that are lower than 4 M⊙, although if the comparison is made with Ne vs. O abundances, the initial masses should have been lower than 2 M⊙. It appears that these models of stars of 2-3 M⊙ are producing too much 22Ne in the stellar surface at the end of the AGB. On the other hand, the comparison with another set of stellar evolution models with a different treatment of convection and on the assumptions about the overshoot of the convective core during the core H-burning phase, provided there is reasonable agreement between the observed and predicted N/O and Ne/H ratios if initial masses of more massive stars are about 4 M⊙.