As we age, our noses do tend to change shape and get bigger. Although the human nose doesn't actually continue growing with age, the cartilage found around the lower part of the nose weakens over time, causing the tip to droop because of the effects of gravity.
Human cartilage is made up of collagen and elastin fibers that will weaken and break down over time. Similar fibres in the skin also weaken with age. As the cartilage and skin weaken, they can no longer maintain the original shape of the nose, resulting in a drooping nasal tip or what is also known as tip ptosis.