Why did the Nazgul not succeed in Bree and on Weathertop?

[Gordis]

Galendor: The wraiths are ghostly forms wearing black cloth robes. Regular swords might not hurt them - but Aragorn's fiery brand could have caught their robes on fire and burnt them up. Then the wraiths would find themselves completely naked and very far from home - a scary thought for anyone!

In UT “Hunt for the Ring” the nazgul were traveling “unclad and unmounted” through Anorien and Rohan – for 17 days, no less! Thus their “naked” state was no big deal for them – only hindered their interaction with the living. At the Ford, the main problem was that they lost all their horses but one and were somewhat battered in the River, I guess, but still the Witch-King was able to ride to Mordor unclad and invisible ahead of the others:

Only the bodies of 8 horses were discovered; but also the raiment of the Captain. It is probable that the Captain took the one horse that remained (he may have had strength to withdraw it from the flood) and unclad, naked, invisible, rode as swift as he could back to Mordor. At swiftest he could not accomplish that (for his horse at least would need some food and rest, though he needed none) ere November had passed. Marquette MSS 4/2/36 (The Hunt for the Ring) RC p. 262

At Weathertop the main thing that scared the nazgul was the Barrow-Downs sword in Frodo’s hands – the WK nearly got killed himself!

But above all the timid and terrified Bearer had resisted him, had dared to strike at him with an enchanted sword made by his own enemies long ago for his destruction. Narrowly it had missed him. How he had come by it – save in the Barrows of Cardolan. Then he was in some way mightier than the B[arrow]-wight; and he called on Elbereth, a name of terror to the Nazgûl. He was then in league with the High Elves of the Havens.
Escaping a wound that would have been as deadly to him as the Mordor-knife to Frodo (as was proved at the end), he withdrew and hid for a while, out of doubt and fear both of Aragorn and especially of Frodo. But fear of Sauron, and the forces of Sauron's will was the stronger. Marquette MSS 4/2/36 (The Hunt for the Ring)RC p.181

Groin Redbeard: Sauron's power was totally destroyed, along with all of his minions, during the battle of the last alliance, but he slowly came back. Could this be true with the Nazgul? Could they have gradually gained power, but not complete power until Sauron had the Ring again?

The nazgul were not destroyed with Sauron at the end of the Second Age. They simply retreated East with the remaining orcs from Barad-Dur.

There can be no doubt that Sauron, well-informed of the Alliance, had sent out such Orc-troops of the Red Eye as he could spare, to do what they could to harry any forces that attempted to shorten their road by crossing the Mountains. It is unlikely that any news of Sauron's fall had reached them, for he had been straitly besieged in Mordor and all his forces had been destroyed. If any few had escaped, they had fled far to the East with the Ringwraiths. Note (No 20) in UT “Disaster of the Gladden fields"

The nazgul simply stayed somewhere in the East, until TA 1300 when the WK founded Angmar. As for the power of the nazgul, I think they were greatly weakened when Sauron collected their Rings to himself. He needed the Nine rings to control the nazgul in the absence of One. If Sauron regained the One Ring, he would likely give the Nine back to the nazgul, making them stronger. That is my take on this.

[Groin Redbeard]

Originally Posted by Gordis

The nazgul were not destroyed with Sauron at the end of the Second Age. They simply retreated East with the remaining orcs from Barad-Dur.

Yes, I know that much. What I meant was that maybe the Nazgul's strenghth is tied to Sauron's.

[Gordis]

What I meant was that maybe the Nazgul's strenghth is tied to Sauron's.

I don't see how it could be - literally. At the end of SA Sauron was reduced to a mere Spirit (twice!) still the nazgul were quite OK - while the Rings lasted.

The nazgul were tied to the Nine Rings (which Sauron held) - and thus to Sauron. Also through the Nine Rings they were tied to the Master Ring. The quote below also speaks of the Ring:

The Ringwraiths are deadly enemies, but they are only shadows yet of the power and terror they would possess if the Ruling Ring was on their master's hand again.

The ringless Sauron was quite powerful by the time of the War of the Ring, IIRC Toplkien said he was more powerful than Morgoth had been, yet the power of the nazgul (as evidenced in the failed Hunt for the Ring) was quite limited. Eight of them (except the WK) were rather pathetic creatures, when left to their own devices. I think it was because they were ringless and Sauron had no Ruling Ring either.