Did this one plug run 2 cubes? AWG16 are rated to handle 16A if you have top shelf 90C rated wires. 4 cable pairs at 12V is a pretty simple power calculation: 4 x 12V x 16A = 768W
Each cube takes 350W. 2 of these should be lower than the 768W max of those power lines, but you probably didn't take into consideration the plug is rated at 13A. This is why the melting started near the plug, not it any wire kink. Each cube should take a single plug directly from the power supply...with its own plug...this means no sharing. 3 cable pairs x 12V x 13A = 468W. More than enough to handle the 350W required by the cube.
2 pcie plugs would have been a seriously cheap solution. I think the real solution is to remove the crappy plug and hardwire 2x 8AWG lines to bus bars.
Lesson learned: KnC cut corners, but buyer beware. Do the math and save your ass.