It's commonly assumed that it was the Nazgul who took part in the attack on the Prancing Pony, but there's actually no evidence in the books to support this assumption.
There is on the other hand plenty of evidence against it.
First of all, the attack of the Nazgul on Crickhollow took place on the same night, but most likely took place later that night. We know that the attack on Crickhollow took place quite close to dawn:
There was a faint stir in the leaves, and a cock crowed far away. The cold hour before dawn was passing. The figure by the door moved. In the dark without moon or stars a drawn blade gleamed, as if a chill light had been unsheathed. There was a blow, soft but heavy, and the door shuddered.
'Open, in the name of Mordor!' said a voice thin and menacing.
But yet Frodo's sleep was disturbed much earlier that night:
In the early night Frodo woke from deep sleep, suddenly, as if some sound or presence had disturbed him.
And in the morning Frodo wakes at almost the same time as the attack on Crickhollow:
He opened his eyes, and heard a cock crowing lustily in the inn-yard. Strider had drawn the curtains and pushed back the shutters with a clang. The first grey light of day
was in the room, and a cold air was coming through the open window.
So from this it seems obvious that at this time the Nazgul did not know that the Ring had left the Shire.
Next, Strider explicitly states that the Nazgul will not attack the Inn:
'What will happen?' said Merry. 'Will they attack the inn?' 'No, I think not,' said Strider. 'They are not all here yet. And in any case that is not their way. In dark and loneliness they are strongest; they will not openly attack a house where there are lights and many people -not until they are desperate, not while all the long leagues of Eriador still lie before us.'
And makes a prediction for what will happen later on that night:
'But their power is in terror, and already some in Bree are in their clutch. They will drive these wretches to some evil work: Ferny, and some of the strangers, and, maybe, the gatekeeper too.'
Then we know from Unfinished Tales that Khamul (second-in-command of the Ringwraiths and probable leader of the attack on Crickhollow)...
...was the most ready of all the Nazgûl after the Black Captain himself, to perceive the presence of the Ring...
So it seems quite absurd that the other Nazgul (particularly if the Witch-king was with them) didn't detect that the Ring was not present in the Hobbits' original room in the Pony.
Finally, there is much work in History of Middle-earth 7 on the movements of the Nazgul, including the following passage (it should be noted here that Tolkien used letter codes to identify the Nazgul, so "DE" refers to two of them):
DE get in touch with Bill Ferney, and hear of news at the Inn. They fear Trotter, but get Bill Ferney and the Southerner to burgle the Inn and try and get more news, especially of the Ring.
So to answer your question: the attack on the inn wasn't carried out by the Nazgul; it was carried out by Bill Ferny and the Squint-eyed Southerner. At this stage the Nazgul didn't know where the Ring was, and the attack was for the purpose of gathering information.