New bulk MAX phase-based ceramics were synthesized in the Ta-Hf-Al-C and Ta-Nb-Al-C systems. Specifically, (Ta1-x,Hfx)4AlC3 and (Ta1-x,Nbx)4AlC3 stoichiometries with x = 0.05, 0.1, 0.15, 0.2, 0.25 were targeted by reactive hot pressing of Ta2H, HfH2, NbH0.89, Al and C powder mixtures at 1550°C in vacuum. The produced ceramics were characterized in terms of phase composition and microstructure by X-ray diffraction, scanning electron microscopy, electron probe microanalysis and scanning transmission electron microscopy. The investigation confirmed the existence of such M-site solid solutions with low solute concentrations, as predicted by first-principles calculations. These calculations also predicted a linear trend in lattice parameter evolution with increasing Hf concentration, in agreement with the experimental results. In order to increase the low phase purity of the produced ceramics, Sn was added to form (Ta1-2 x,Hfx)4(Al0.5,Sn0.5)C3 and (Ta1-x,Nbx)4(Al0.5,Sn0.5)C3 double solid solutions, thus resulting in a higher content of the 413 MAX phase compounds in the produced ceramics.