I think that the issue with using Lord of the Rings as a comparison for 3.5, is that the standards are all over the place. It seems that all of the comparisons that were linked thus far are skewed to the writer's opinion. So, lets compare by making a list of characters and their feats.
Aragorn: A crazy amount of tracking and survival, several mass combats, where he is credited with at least 6+ kills(moria), intimidating Sauron.
Legolas: HELMS DEEP. The contest with Gimli.
Gimli: See above.
Ok, so given the fact that the Urak-Hai are more than typical orcs, I would stat them as similar to Bugbears. Killing 6 in a fight, with out any real injury means that the CR should have been well below the level of the party. So, Aragorn killing 6 of them would be a CR 11 encounter I believe.
Gimli and Legolas demolish a large number as well. I seriously do not see how they would be 3-5th level.
At 3-5th level, the 20ish orcs in Moria + Cave Troll, should have definately killed someone. While you can claim DM fiat, that is dodging the issue. Not to mention the mass combats. Boromir takes out how many Uruk-Hai at the river?
Overall this is really a fruitless debate, 3.5 can be played as High Fantasy, but it requires the DM and the Players to be on board and make appropriate choices.
Also, how would you stat people like Beorn and Tom Bombadil? Clearly Tom was quite powerful by himself.
The uruk-hai bit smacks a bit of "clearly LotR orcs are more badass because they threatened these badass characters, and clearly these characters are high-level because they smacked down badass super-orcs." Pegging uruk-hai as bugbears is a bit arbitrary; you can make an equally good case for them being half-orcs with PC class levels (at least part of their being hardcore is supposed to be because of their elite training) while ordinary orc soldiers are level 1 orc warriors.
D&D trolls are not Tolkien trolls, they're Three Hearts & Three Lions trolls. Tolkien trolls are more similar to D&D ogres in depiction than they are to D&D trolls. An ogre and a swarm of orcs is a noteworthy fight for a level 5 party but not an inevitably fatal one.
Beorn and Tom Bombadil are NPCs with non-PC-available abilities. Tom in particular is nonhuman and probably has extensive abilities owing to whatever kind of bizarre fey he is.
See, there in lies the rub. There are plenty of reasons to believe that the characters could be anything you want. Truthfully though, everything I have seen points to them being higher than 5th level. The mass combats alone make me think that they would have to be higher level than that. Now, I get the desire to paint them as closer to average, but I just don't see why. As someone pointed out, the saves that the characters made alone suggest that they are higher level. So, they made some saves that really wouldn't be possible for lower level characters, they fought in massive battles killing a ridiculous number of foes without taking any substantial loss, and the manner in which they are described suggests they are more than low level. In a world where there are great dragons, liches, death knights, etc. it makes sense that the heroes that oppose them are more than low level. I also think that this basically boils down to the type of game you want. It is not impossible to play a high fantasy game at high level. It depends on the players and the DM. If you want a high fantasy game, you can run one, but you can also do it over 20 levels. If players want to twink out and make high powered characters, this is more difficult. Honestly, none of the characters from LotR were particularly optimized characters.
Aragorn would be some combination of Fighter/Ranger/Paladin, Legolas some type of Fighter most likely, focusing on bows. Gimli, the same, a fighter with an axe. Nothing special. Gandalf appeared to be a mage with a proclivity for Fire magic, so perhaps some type of an Evocation Specialist probably barring necromancy and something else, I would really have to look back at the book to figure out what, though I am leaning toward Transmutation as Illusion, Enchantment and Evocation seem to sum up his spell selection. The Hobbits, well Sam was a Fighter, Pippen was a rogue, Merry a fighter/rogue, Frodo is the most difficult. He didn't seem particularly gifted at combat, he didn't possess magic, he basically just walked. I guess we could call him a monk, he certainly made good saves, and they are good at that I suppose.
Honestly, there is no reason that with these kind of characters, built for story rather than Uberness, that you could not keep a High Fantasy feel going through all twenty levels.