Use of the Morgul Knife Was Key
Jumper

I have read the postings on why the Nazgul failed to follow up their attack at Weathertop and finish off the party and/or seize the ring from Frodo with interest. This is something I have often thought about. Personnally, I believe that the Nazgul thought their best chance was to use the Morgul knife and let Frodo submit to them through it.

Consider their situation -- they had encountered and battled Gandalf just days before, Gandalf's whereabouts were unknown to them and he could return at any time, the hobbits and Aragorn had put up more of a fight then expected and finally, they were not to reveal themselves per Sauron's orders. Their plan would have worked except for the medical attention of Aragorn and Glorfindel enroute to Rivendell and the extraordinary strength of Frodo's constitution.