My Pixhawk boots fine when powered by the power module only, but when I connect 5V to the servo/motor output rail, it doesn't boot but just makes clicking sounds instead.
Video demonstrating the problem:
The 5V being supplied to the servo output rail comes from a J&K Universal Power Distribution board with built in 5V and 12V regulators. I measured the 5V regulator output to be 5.00V exactly on my voltmeter.
If I connect the 5V to the servo output rail after booting, then it works fine too.
My wild guess is that it's a timing issue at power up, e.g. if one of the power inputs get's power a before the other, then the Pixhawk freaks out.
I tested it with a different 5V regulator afterwards and then it did boot. But that doesn't explain why it doesn't boot with the power distribution board's regulator.