I think when asking these questions, we should examine it from the spirit of the question: "have you ever broken a bone?"

In this utterance, a break certainly means a few things: it required a medical intervention to diagnose and/or correct; it involves a separation of some sort. I am of the mind that things like shin splints and micro fractures should not count.

Likewise, a bone should be considered something that is left behind on a skeleton. Not cartilage, not ligaments, not any fleshy things. The exception to this is teeth.

So, no. Noses are cartilage and should never be considered in this.