Can blonde box dye lighten dark brown hair?

Yes, but you can't necessarily guess or control the outcome.  All hair dye has both a dye (color) and come chemicals that open the cortex of the hair and remove some of the color. That is why a person can still get a dye line when they color their hair the "same" color as their natural color.

The issue you may run into is that the hair that is growing out will react differently to the color than hair that has already been colored before.  

My hair is brown, and I've always just dyed it light ash blonde (or some other shade close to it) without any problem.  Platinum is really severe, and I don't think that's a necessary step - but I'm not a professional.