Is It Possible for Skin to Not Reattach to Cartilage After Rhinoplasty?

What you describe is not a problem of the skin "reattaching" to the cartilage after your rhinoplasty but that it sounds like you don't have enough cartilage structure to support the weight or amount of nasal skin you have.   Over-reduction of the natural nasal cartilages in rhinoplasty instead of reshaping them can lead to this problem.   It is definitely correctable with revision rhinoplasty.  Rebuilding some of  the lost nasal shape with cartilage grafts from the septum, ear, rib or cadaveric rib cartilage can restore structure to the middle third, upper lateral cartilages and nasal tip and nostril support to achieve a functional and attractive nose. 

The key to both primary and revision rhinoplasty is support. The bony and cartilage structure that supports the nose needs to be maintained. Depending on how strong your cartilage is, more or less can be removed in rhinoplasty and not compromise breathing. Other ways to make a nasal tip smaller without removing a lot of cartilage is by the use of grafts of your own cartilage. I find that I use grafts of the patient's own cartilage in over 905 of primary rhinoplasties and almost 100% of revision nosejobs.

Although it is hard to tell without seeing your pictures what you are describing is likely just related to over-resection of your cartilages resulting in poor structural support and collapse - this can be improved with a revision