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We demonstrate that the residual carbon concentration in the drift region can have a significant impact on the reverse leakage, breakdown voltage, and breakdown stability of GaN-on-GaN vertical diodes. Two generations (Gen1, Gen2) of polarization-doped p-n junctions with different C concentrations were compared, in terms of avalanche voltage, avalanche instability, and deep-level concentration. The original results collected within this paper show that: 1) both generations of devices can safely reach the avalanche regime; diodes with a lower residual CN have a higher reverse leakage and a lower avalanche voltage, due to an uneven distribution of the electric field; 2) the presence of residual carbon can lead to breakdown walkout, i.e. a recoverable increase in breakdown voltage under reverse-bias stress. Specifically, devices with higher C concentration show a fully-recoverable breakdown walk-out, whereas the breakdown voltage is stable in devices with lowerC concentration;and 3) steady-state photocapacitance measurements confirm the presence of CN in both generations, and are used to assess the relative difference in concentration between Gen1 and Gen2, even for levels below secondary ion mass spectroscopy (SIMS) sensitivity. The results described in this paper indicate the existence of a trade-off between breakdown voltage (increasing by improving compensation) and breakdown stability (improving by reducing CN concentration) and are of fundamental importance for the optimization of GaN power devices.