Is It True That the Tip of the Nose Falls if You Remove a Bump on the Bridge?

That is incorrect.  When removing a bump on the bridge, the columella tends to bow downwards and give an illusion of a lengthened nose.  After removal of the hump, attention is directed to reduce any columellar show to keep the nose within proportion.  Columellar show is a telltale sign of having had a rhinoplasty.  Simple hump removal can be done quite easily through a closed rhinoplasty without any elongation of the nasal tip.  Open rhinoplasty will extend the healing time with more swelling, bruising, and has other associated complications.  

Three things can happen after removing a nasal hump. The tip can remain in the same position which is most common if the surgeon does not affect the supporting mechansims of the tip of the nose. Second, the tip can rotate upward if the hump reduction goes all the way down to the tip and includes the cartilage septum at the anterior septal angle. Third, the tip can only fall if the supporting attachments to the septum and the upper lateral cartilages are disrupted allowing the tip to fall. So the bottom line is the tip position is dependent upon what the the surgeon intends to do with your nose not the hump reduction itself.

In general, taking down a bump on the nasal bridge shouldn't cause the tip to fall. Depending on the surgical approach, though, the tip can become less stable and require extra support to maintain it's current position.

One thing to consider with removing a bump on the bridge is the matter of preventing pinching of the middle third of the nose as the nose heals. Spreader grafts can be placed in this case to prevent this from occuring.