If taking out the cap doesn't solve the problem, it does not seem to be related to powering the voltage regulator (?).
It takes about 600µs until the 3.3V and 5V are fully on the 3.3V and 5V pins (> 30 sec off). It takes about 400µs when I reconnect the power supply again after disconnecting it (< 5 sec off).
Maybe the SAM3X tries to do some operations when the supply unit isn't fully stable, at around 500µs or so. Just a guess.
It's very annoying, I need the DUE to boot directly and correcty after it is switched on. It is used in an embedded project...