Can you break or crack your nose cartilage?

Thank you for your question! An impact to your nose is the most common cause of a nasal fracture. Your doctor can usually diagnose a broken nose by performing a physical examination. Since this will include touching your face and nose, your doctor may apply a local anesthetic to numb your nose before examination. I recommend consulting with an experienced board certified plastic surgeon. Good luck!

Septal cartilage fractures are possible and usually associated with less bleeding or bruising. The immediate point to be careful about is a septal haematoma as it may require evacuation to avoid damaging the septal cartilage further. This is the main value of a visit to an emergency department straight after the injury.
It is normal for the bridge to be sensitive for a little while after the injury and it can reflect also a small fracture with or without displacement over the nasal bridge.
Ultimately, the pain and tenderness will settle and the reason for surgery would need to be to manage problems related to breathing or external shape, or a combination of the two.

Yes. It is very possible to "fracture" the cartilage of the nose. Surgery is typically indicated if there is either a cosmetic deformity (ie. a change to the external appearance of your nose) or a functional issue such as difficulty breathing.  As swelling from a nasal injury can obscure the actual deformity if present or make breathing worse, it is often important to wait a period of time to evaluate whether there is indeed something that needs correction.  You should have your nose looked at by surgeon experienced at performing nasal surgery. Good luck.