First, I haven't found any indication that another game does this.
Having said that, I'd love to try it, or find examples of it, and was looking for examples when I came across this question (hence the (somewhat) necro - but an answer hasn't been accepted, so I might as well give one).
I believe such a method would be useful, though its value would vary by genre or specific game mechanics.
A first person shooter would have little use for this, since player-player interaction and fluidity of experience are quite far removed from each other, and thus pure client-server seems to suffice. What I mean by that is: player movement is rarely inhibited by other players, and player interaction is rarely more complex than shooting (which usually will just lower their opponent's health in the case of a hit, and the opponent's health is rarely visible to the player). Often, their only awareness of lag is usually taking damage a little after getting behind cover, or dying after riddling an opponent with bullets, but those are both critical decisions that should be left to the server.
Movement, which you mentioned in a comment on another answer, is probably the best candidate for this to work. If your particular subgenre of FPS allows players to side-step quickly, zigzag, bunny-hop, or puts any value into other quick movements, some extra predictability can help players hit their target, especially with powerful weapons with a low rate of fire. At the same time, it's also a great avenue for cheating. Sure, the server will correct it, but making movement even more sporadic due to invalid information and frequent corrections from the server would certainly give me a harder time shooting you ;)
Since you tagged your question first-person-shooter, the answer could be left at that. But since the same question without that tag would likely be considered a duplicate of this, I hope you don't mind me expanding to other genres.
Other genres with more direct interaction could benefit greatly from this model as far as the end user is concerned, even though the simulation on the server is unchanged.
A brawler requires players to read each other's movements and react to each other's actions before they take impact. If a character has a 400ms power-punch animation but only sees it begin 200ms after their opponent has started the action, they're in trouble. But if the opponent's client can say "Hey, I've told the server about this action; you should start this animation while the server confirms it for you", the player could see the animation begin much sooner and have time to react, or at least be less perplexed that their character's ragdolling across the stage half a second later.
Maybe. In ideal circumstances. Worst case is more traffic and no improvement, but it won't make the experience worse.
A more physics-driven game, like a driving game, depends quite a bit on plausible motion. Being notified of a car turning and taking that into account in the client-side prediction would result in less correction when the authoritative server information comes through. Or a quick boost before a car rams into you and knocks you flying will be less jarring if you see them boost just before they hit you, rather than just after.
In the end, the server simulation should be unaffected, and there are new cheating avenues to try and take care of. But these cues could make lag less jarring for your typical player, who generally has no idea why lag happens.