When does your nose tip start to weaken and droop? Around what age?

This is a very individualized process that depends on a number of factors most of which are genetically determined:  your cartilage strength and thickness of the skin are the two most important.  Likely by early to mid 30s there is some weakening of the supporting ligaments between the cartilage that results in early tip droop.

As we age, the nasal anatomy changes and the tip may start to droop. In addition I have seen patients nasal anatomy change becoming twisted or develop a dorsal hump. These features may occur as early as your 40s or 50s. Over time they become more exaggerated.

The droop associated with aging is related mostly to the skin changes. The cartilage most likely does not contribute to what sometimes occurs but not always in the nose during the later years. There is no known way to predict who will have significant nasal tip drop with age. Family history may be helpful in some instances. Having a rhinoplasty early in life- in my opinion- can help delay or prevent nasal tip droop in later years.