If the broken tile was due to an accident, then chances are good that the substrate that the tile was attached to is still in good shape, and you can proceed with a simple replacement to complete your tile chip repair. Other tile cracks and failures can be due to the collapse of the "backer" or substrate. I have seen this often in bathrooms where water plays its dirty tricks and rots away the drywall or sub-floor behind the tile. For these issues, you might want to just go fishing, since you've opened a "can-of-worms." That type of cracked tile repair project often needs large scale or expert help.