It seems, due SAM-BA (SAM Boot Assistant) inside ATSAM3X8E being listening to both Native USB and UART USB ports after a Reset for programming, allowing talking with SAM-BA, this time from your PC, through BOSSA (bossac.exe, Basic Open Source Sam boot Assistant), then it would make not possible to program an Arduino Due while having both USB port connected to a PC.
Can somebody confirm, this is the reason?