Why won't the bump on my nose flatten when I press it with my fingers (isn't it cartilage)?

The bone and cartilage of your nose is what gives it it's shape and contour.  You can't change it by simply pushing on it, as it is too strong.  Your best bet is to see a board certified plastic surgeon and discuss how best to correct this issue which is bothering you so much.  Knowing what is entailed can ease your mind.

While cartilage is softer than bone, it will still not compress the way your are hoping it would.  Surgery is the only method, in my opinion, to address this concern.  Good luck and be safe.John T. Nguyen, MD, FACS, FICSDouble Board Certified Plastic Surgeon

Unfortunately the safest most predictable way to remove a dorsal hump (which is often bone + cartilage) is through rhinoplasty surgery.  Pressing on the convexity will not change it.  Nonsurgical rhinoplasty with fillers can camouflage it, but is not permanent.  

It is hard to figure out how bad the bump is with out a photo but it sounds like it is a permanent part of your nose.  The bone and cartilage structures can not be molded with finger massage.  It requires surgery to change and adjust the size of the cartilage in your nose.  Sometimes injections can be used to hide the bump.An in person evaluation will be more valuable.Best WishesDr. Peterson

It is probably a combination of cartilage and bone. neither goes away without surgery. But you would be a very poor candidate for surgery based on your reasons. There are hundreds of thousands of women in this country alone with a hump on their nose that are successful in life and in relationships. It isn't the hump that is your problem.