'Your quest is known to us,' said Galadriel, looking at Frodo. `But we will not here speak of it more openly. Yet not in vain will it prove, maybe, that you came to this land seeking aid, as Gandalf himself plainly purposed. For the Lord of the Galadhrim is accounted the wisest of the Elves of Middle-earth, and a giver of gifts beyond the power of kings. He has dwelt in the West since the days of dawn, and I have dwelt with him years uncounted; for ere the fall of Nargothrond or Gondolin I passed over the mountains, and together through ages of the world we have fought the long defeat.
('I it was who first summoned the White Council. And if my designs had not gone amiss, it would have been governed by Gandalf the Grey, and then mayhap things would have gone otherwise. But even now there is hope left. I will not give you counsel, saying do this, or do that. For not in doing or contriving, nor in choosing between this course and another, can I avail; but only in knowing what was and is, and in part also what shall be. But this I will say to you: your Quest stands upon the edge of a knife. Stray but a little and it will fail, to the ruin of all. Yet hope remains while all the Company is true.')
Yet in the next chapter, Farewell to Lórien, one can find something that seems to indicate that there was something unconventional, a sad element in their relationship:
Now Galadriel rose from the grass, and taking a cup from one of her maidens she filled it with white mead and gave it to Celeborn.
'Now it is time to drink the cup of farewell,' she said. `Drink, Lord of the Galadhrim! And let not your heart be sad though night must follow noon, and already our evening draweth nigh.'
One might argue that here Galadriel is speaking generally about the fate of Elves in Middle-earth, yet a similar indication can be found in the chapter Many Partings in ROTK :
Then Aragorn took leave of Celeborn and Galadriel; and the Lady said to him: 'Elfstone, through darkness you have come to your hope, and have now all your desire. Use well the days!'
But Celeborn said: 'Kinsman, farewell! May your doom be other than mine, and your treasure remain with you to the end!'
Thus, there is something about their relationship that has always puzzled me. Why is it that both Galadriel & Celeborn indicate that they might be separated in the future, and what are the reasons for this?
I am not much accustomed with Elven marriage, yet I take it that though Galadriel & Celeborn were married in Beleriand/Middle-earth, they were not yet "officially" married in the eyes of the Eldar - since Galadriel was born in Valinor under the Light of the Trees, it might be that their marriage would only become official with Eärwen's and Finarfin's consent (indeed, Finarfin did not join the Rebellion but stayed in Tirion).
One must not forget that Galadriel was the only one of the House of Finwë who married someone not of the Eldar (provided, of course, that one follows the story as narrated in the Silmarillion).
But at any rate, even if the above is true, then why would Celeborn speak about a 'doom'? After all, him having married Galadriel without her parents' permissions seems hardly important - in fact, if the parents are absent, their consent is given by others (Melian & Elu Thingol, for instance), so this should hardly pose a problem.
Clearly, the reason for why Galadriel & Celeborn parted at the end of the Third Age -Galadriel left with Elrond, Gandalf and the other Ringbearers, whilst Celeborn still lingered in Middle-earth, first in his newly founded Woodland Realm, and later in Rivendell), should be something entirely different... something on which Tolkien unfortunately didn't expand in any of his writings for I think it is a particulary puzzling concept.
Due to lack of further information, one might indeed conclude that Celeborn could not yet part together with is beloved spouse before Galadriel had obtained her consent from her parents - which seems quite an insipid reason at best.