I think you might have overwhelmed him with that response, Doc 
Reading the application note for the MBI6651 you're correct that the cathode side of the LED cannot be treated as GND. dostjusting, the DIM connector would need to be connected all the way to the other side of the board to the GND connector to turn the LED off. Oddly enough the appnote doesn't even mention what happens when you leave the DIM pin floating.
If you're using an external power supply for your LED as the schematic describes and you're trying to control the DIM pin with an Arduino, make sure that you have a wire connecting the GND on the power supply to the GND on the Arduino.