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

[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.