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Context. The planetary nebula luminosity function (PNLF) is a standard candle that comprises a key rung on the extragalactic distance ladder. The method is based on the empirical evidence that the luminosity function of planetary nebulae (PNe) in the [O III] λ5007 nebular emission line reaches a maximum value that is approximately invariant with population age, metallicity, or host galaxy type. However, the presence of bright PNe in old stellar populations is not easily explained by single-star evolutionary models. Aims. To gain information about the progenitors of PNe at the tip of the PNLF, we obtained the deepest existing spectra of a sample of PNe in the galaxy M 31 to determine their physico-chemical properties and infer the post-asymptotic giant branch (AGB) masses of their central stars (CSs). Precise chemical abundances allow us to confront the theoretical yields for AGB stellar masses and metallicities expected at the bright end of the PNLF. Central star masses of the sampled PNe provide direct information on the controversial origin of the universal cutoff of the PNLF. Methods. Using the OSIRIS instrument at the 10.4 m Gran Telescopio Canarias (GTC), optical spectra of nine bright M 31 PNe were obtained: four of them at the tip of the PNLF, and the other five some 0.5 mag fainter. A control sample of 21 PNe with previous GTC spectra from the literature is also included. We analyze their physical properties and chemical abundances (He, N, O, Ar, Ne, and S), searching for relevant differences between bright PNe and the control samples. The CS masses are estimated with Cloudy modeling using the most recent evolutionary tracks. Results. The studied PNe show a remarkable uniformity in all their nebular properties, and the brightest PNe show relatively large electron densities. Stellar characteristics also span a narrow range: ⟨L*/L⊙⟩ = 4300 ± 310, ⟨Teff⟩ = 122 000 ± 10 600 K for the CSs of the four brightest PNe, and ⟨L*/L⊙⟩ = 3300 ± 370, ⟨Teff⟩ = 135 000 ± 26 000 K for those in the control set. This groups all the brightest PNe at the location of maximum temperature in the post-AGB tracks for stars with initial masses Mi = 1.5 M⊙. Conclusions. These figures provide robust observational constraints for the stellar progenitors that produce the PNLF cutoff in a star-forming galaxy such as M 31, where a large range of initial masses is in principle available. Inconsistency is found, however, in the computed N/O abundance ratios of five nebulae, which are 1.5 to 3 times larger than predicted by the existing nucleosynthesis models for stars of these masses.