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

Generally, the tip does not need to be elongated following removal of the nasal hump.  It does however need to be re-supported in order to prevent it from drooping over the course of healing.  Rhinoplasty can weaken the support of the tip so regardless of the maneuvers performed, the support needs to be re-established. This is not the same as making the nose longer.  As mentioned earlier, bringing the hump down, the nose will appear longer.  This can be modified using other maneuvers to make the nose look balanced. 

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.