I've stayed out of this thread for a while, but since Bullgrit want's some actual experiences, here are mine. These are my observations.
1. D&D (even 3.5) was never designed to mesh with a skill system.
There are too many effects that are mundane dice rolls (swim, open locks, hide, heal checks) that magic doesn't aid, it eliminates (freedom of movement, knock, invisibility, neutralize poison/remove disease). This means that skill use is always second class to magic, provided magic is available. add to the fact there are cheap ways of replicating said effects (scrolls, wands, potions) and you have a game where skill users are almost always inferior except when a.) the magical effect is unavailable or b.) required multiple times in succession without rest.
Note all these effects are low level too, meaning when the casters have graduated their attack and defense magic to higher level slots (for more damage, better effects and higher DCs), what better way to fill unused spell slots than with "problem solver" magic? The only time I didn't see mages doing this is when they "purposefully" avoided doing so to give skill users "something to do" (which, even when meant in earnest, still comes off patronizing).
2.) More Casters, Worse the Effect
If you're "blessed" to have a game consisting of one warrior, one mage, one priest, one expert (typically ftr/wiz/clr/rog) the effect is less pronounced than if you have any duplicates or supplemental casters (such as sorcerers, druids, or bards). The more players who turn to casting, the worse it is for the non-caster. It multiplies the number of spells available, adds to the spells known pool, and allows the casters to split the burden of their primary role (nuker, healer) to augment into other roles. For example, one mage could focus on attack magic, while another focuses on summons and divination, and a third on buffing and transport (true story, saw it happen).
3.) The Games ever-increasing need for magic
As the game goes higher and spell slots become more prevalent, magic becomes the go-to for challenges. Why cross a rickety old bridge when fly gets you there safer, or teleport moves the whole party? Additionally, non-casters find themselves heavily dependent on magical items and buffs to remain effective, while casters need very little beyond defensive items. This leads to two phenomenon; the first is the fact casters become the primary "solvers" of problems outside combat (like transport, info gathering, even camping) which limits the screen time of non-casters and secondly fighters become extremely dependent on their wizard and cleric buddies to "keep up with the Jones" among fighting level-appropriate monsters.
4.) Versatility is only a Day Away
This one primarily applies to casters with prep spells; spontaneous classes actually are in the same boat as non-casters (and hence are typically viewed as weaker than prep-casters). Fighters, rogues, etc make relevant character choices twice; at char-gen and at level-up. Here, the select feats, skills, class-abilities etc that define their existence. If a particular adventure shows signs of requiring different expertise (A tomb of traps and undead, for example), these classes cannot adapt, they are fixed with those choice (the ranger cannot change FE:Giants to Undead, the rogue can't reinvest those ranks in Diplomacy into search, the fighter can't trade Imp. Critical: Scimitar for Power Attack, etc).
Whereas the cleric, druid, or wizard can TOTALLY remake thier PCs primary function with 24 hours. Tomb of the Undead? the cleric memorized Death Ward and Searing Light while the Wizard trades Sleep for Magic Missile. This means a caster, with an opportunity to rest, can become dominant in a given situation while a choice the non-caster made at level-up comes back to bite them.
5.) Realistic Limits on "Unlimited" Powers
This one goes back to concepts of "how many locked doors ARE there, anyway?" Most adventuring parties, with no time limit and ample opportunity, prefer to rest when their mage and cleric are low on magic (for reasons that should be obvious, but if not, see # 3). That means fighters "rarely" make many more "attacks" than a caster can. This doesn't mean he and the wizard have a 1:1 attack:spell ratio, but unless the fights are sufficiently "easy" the the wizard doesn't feel he need to use a lot of his spells in a given fight, the fighter won't have a lot of opportunity to "wipe out" all foes while the wizard stands idle. Sadly, the effects of magic has rendered the opposite true from time to time; the fighter stands idle (because of magical effect or simply low initiative) and the wizard wins the fight.
In Conclusion
A fighter is still a powerful class, it is NOT however a versatile one. That is whee casters, shine, versatility. There is rarely a moment a wizard doesn't shine; he can sling fireballs, teleport the party back home, divine lost info, or sneak past a guard undetected. Non caster do not have that versatility and where they are supposed to shine (combat, skills) the wizard can match or upstage them.
Simply put, a caster can do "everything" a non-caster can do, and a lot of things they can't.
That's why they don't matter after 11th level; they stop bringing something unique to the table that can't be replicated with the right spell or item.