There appears to be some significance surrounding these numbers throughout Tolkien's 'legendarium'. Some of this is evident in the old rhyme Gandalf mutters to himself as he rides with Pippin to Gondor:

"Tall Ships and Tall Kings, three times three
What brought they from the foundered land, over the flowing sea?
Seven Stars and Seven Stones and One White Tree

The times which each of these numbers crop up is almost alarming! Take a look:

» 3 Elven Rings
» 3 Unions of the Eldar and the Edain
» 3 Hallowed Jewels (the Silmarils)
» 3 Hairs given by Galadriel to Gimli
» 3 Lords of Gondor and Arnor (Elendil, Isildur, Anarion)
» 3 Volumes in LOTR - this is purely coincidental, and was only done in this way due to the general shortages of paper and book-binding materials following the second world war.
» 3 (major) ring-bearers - Bilbo, Frodo and Sam. This is perhaps tenuous, as Gollum also bore the ring, and for longer than the other three put together. Bombadil also handled the ring, but can't really be classed as a 'bearer'.
» After the breaking of the fellowship, they split into three groups: The Three Hunters (Aragorn, Gimli and Legolas); Frodo and Sam, who met up with Gollum; and Merry and Pippin, who met up with Fangorn (Treebeard).
» Three cities of Gondor: Minas Anor, Minas Ithil and Osgiliath
» The Free Peoples: Edain, Eldar and the Dwarves (there were others - Hobbits, Ents etc - but generally the three are the only ones mentioned)
» 3 peaks above Moria - Barazinbar, Zirakzigil and Bundushathur
» 7 Dwarven Rings
» 7 Stars on the Banner of Elendil
» 7 Palantiri
» The light of Laurelin and Telperion, the two Trees of Valinor, lasted for 7 hours
» 7 circles of the cities of both Minas Tirith and Gondolin
» The last Dwarven king called Durin was the seventh of that name
» 9 members of The Fellowship
» 9 Rings for Mortal Men, which led to 9 Nazgul
» 9 Ships escaped the ruin of Númenor
» One Ruling Ring
» One White Tree (the Tree of Gondor)
» One jewel in the crown of Elessar (and indeed in the original Elendilmir, recovered from Isengard by Elessar in the early Fourth Age

Admittedly there is rather less symbolism surrounding number nine - or so it initially seems. A note I spotted over at Mythictruth.com while researching this very subject had some interesting things to say about nine:

[Direct quote]:
Also of note are the Nine Ships [three times three] that escaped from the destruction of Númenor and the subsequent appearance there after of the Nine Númenorean Kings who were seduced by Sauron and eventually became the Nine Riders. It is almost as if Tolkien is saying that upon each ship there was one who betrayed them.
[End of direct quote]