I have Arduino Due and PCA9685.
I use this library: https://github.com/adafruit/Adafruit-PWM-Servo-Driver-Library
The board does not do anything with servo connected. I tried to use SDA/SCL pins and SDA1/SCL1 pins, I tried to use or not use pullup resistors. The board does not react in any case.
The PCA9685 is 100% working, because it operates properly when I connect it to my Arduino Mega using the same library, it works without pullups.
Also I noticed the following indicium, when I tried to use this scanner http://playground.arduino.cc/Main/I2cScanner I have no detected devices on SCL1/SDA1 but have detected 2 devices at 0x40 and 0x70 when I connect the board SDA/SCL pins.