It would be a lot more common to burn out barrels than it would be to shoot down a bomber, let alone two...
As has been said, all rifled barrels wear out. The rate at which they wear depends on the power of the ammunition (high-velocity rounds use more propellant, generating more pressure and heat, so more wear). It also depends on the details of the design of the shell and rifling. But most importantly, it depends on how fast the gun fires and for how long.
It doesn't take much firing to heat up a barrel, and the more rounds are put through it per minute the faster it heats up. Also, the thinner the barrel the faster it heats up. And the more it heats, the faster it wears. To give one example, the .50 Browning M2HB (heavy barrel) fires at around 500 rpm (ground use) but the maximum continuous rate of fire to avoid oveheating is 40 rpm. In the lighter-barrel aircraft version of the gun, 30 seconds continuous firing would ruin the barrel.
In AA use, authomatic weapons (especially naval ones) were often given water-cooled barrels, which avoided this problem.
Tony Williams: Military gun and ammunition
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My questions are 1; How frequent was "burning out barrels"? Especially against kamakasis' in the Pacific. 2; Were there any specific number of rounds you were expected to use to knock a plane down? 
3; Was there problems with artillery pieces wearing their grooves out?
