Yes, you can break a diamond. Hitting a diamond with a hard blow from a hammer, for instance, is liable to crack, split or outright shatter the diamond.
The idea that diamonds are indestructible comes from the misconception of the word "hardest" when it comes to discussions of diamonds. Hardness denotes how difficult it is to scratch the material. Tenacity is used to describe how difficult the material is to break. While diamonds are rated a 10 on the MOHR scale of hardness, they are only rated fair when it comes to tenacity. This is why diamonds can still chip easily on the edges.