It's not about server-side computation resources or bandwidth, it's about latency. If it takes 50ms to communicate your position to the server and 50ms for them to reply, that's 1/10 of a second for every position update, which is an eternity in high-level play. Netcode is really hard, anti-cheat is really hard, and mashing them together is really, really hard :)
I think it could be possible if you structured your FPS like an MMO: build the game out of "zones" with "rooms" in them, and then clients only get pushed events for the "zone" and the "room" they're in. (Unlike an MMO, the whole map and all relevant models would always stay loaded into memory, so there wouldn't be any latency in transitioning between rooms; there'd just be a moment where you'd be outside and not see anybody, before they popped in from your perspective and you popped in from their perspective. You could cover it up with a lens flare or something.)
Of course, people would then gyrate back and forth across zone/room thresholds as a way of protecting themselves, which seems obnoxious, but also seems like a very easy behavior for the server to detect and punish.
The top-down MMOs I've played have line-of-sight fog-of-war while in caves/buildings (with limited range smaller than the screen in caves), which has the same issue. The clients have access to invisible things so they can accurately predict when they move - including when they become visible - and correct a split second later if the prediction was wrong.
This is already how some first-person shooters work, notably Counterstrike. Maps are segmented into areas and you only get updates for things happening in the local area. This has actually been a source of bugs, where a grenade going off near the area never gets its sound transmitted to people in the area next to it as to those clients that grenade was never thrown.
Funny enough this was actually an issue in the earlier days of a hardcore bigworld MMO I used to play called Mortal Online. Reds (gankers) would hang out near the boundaries and pop in out of nowhere. This was largely more of an engine limitation of UE3 as I understand it, and was eventually mostly worked around. Now many of the UE3 painpoints are softened with UE4 they have started Mortal Online II.
