Is it possible to use a DRV8825 to drive higher current stepper motors?
You will burn out the DRV8825 if you use it to drive higher rated stepper motor.

Majorly, the reason being integrated H bridge and that takes major real estate on the silicon die. So, if you overload it, it might just shutdown due to temperature or over current(RSense). 

If you want to actuate a stepper motor larger. I would go with something which has an external H bridge arrangement. Like a stepper indexer.

You can look TI's DRV8711 for this application. It caters these kind of applications and provides more functionality than DRV8825.