Any regulator, linear or switching, has a "dropout voltage". This is the minimum input voltage required to get the rated output voltage, and can change a little as current increases.
From previous testing, the CC BEC needs around 2v higher input than the output. So, for 6v output, you should have at least 8v input. If the input drops, then the output drops too. The CC BEC I tested completely shuts down when the input gets to ~4.5v.
This is the results of the test I did a while back:
