Will Nose Bump Become Larger or More Pronounced As We Age?

Generally speaking, no, a bump on the bridge of the nose won't grow with age. Insurance generally covers procedures that are to improve function and your health, but not those that are just for aesthetic purposes. Therefore, smoothing down a bump probably won't be covered by insurance.

 As we get older the nose appears to grow larger. What is really happening is that we are losing fat in the nose and have loss of support of the tip along with resorption of bone. The tip drops as well and the nose appears larger.

I assume you are talking about the dorsal hump with is one of the most common reasons someone may chose to have a rhinoplasty. This is most often inherited from a family member and is seen more often in certain ethnicity. It will grow to certain size through puberty and into the early 20's sometimes but then it should remain unchanged. It is essentially a cosmetic procedure unless there is an obvious deviation that affects breathing. If so it can sometimes be covered by insurance but typically it is a cosmetic procedure.

Although certain aspects of the nose change with advanced age, the hump at the bridge of one's nose is comprised of nasal bone and cartilage which do not grow. The support structures of the nose, however, can relax with time, causing the nasal tip to droop and this can make the hump look more pronounced.
Insurance reimbursement is often available for Rhinoplasty when the only goal is to breathe better, aka "functional rhinoplasty."