The bane of a high speed gamer.
This bugbear of gaming appeared early. I pull out my tattered old D&D Red Box...Save vs Rods or Breath, Save vs Poison or Death Ray, Save vs Staves or Spells.
We move through several generations of games..
I move on to a generic product from the Palladium lineage, Heroes Unlimited...Save vs Coma/Death, Save vs Magic...my favourite nonsensical saving throw...Pull/Roll with Punch/Fall/Impact, and dozens of others to cover resisting any type of effect that could possibly hinder a character.
Lets move onward to advanced games where storytelling is more important than mechanics (or so was the claim)...I'll pull out the more socially oriented of White Wolfs original World of Darkness, Vampire the Masquerade (I could have pulled out the more combat focused Werewolf, but we expect more detail in its combat resolution mechanisms)...even in this combat is divided into a roll for attack, a roll for damage, then a roll to soak (and hopefully avoid a chunk of that damage). Most actions in the game series follow a similar procedure...aggressor rolls a bunch of dice to claim successes (often with a difficulty based on the situation at hand), while the defendant rolls a bunch of dice to negate those successes.
A quick detour over to wargames...the common conflict pattern in these games follows the same structure...roll to see if it happens, roll to see how bad it is, then roll to see how much of it you can avoid.
Attack, Damage, Armour Save.
As originally intended, saving throws were designed to allow characters a chance of avoiding things that might otherwise simply happen automatically. In old D&D, spells just happen; without saving throws, there would be no way to avoid the incoming sleep spell or magic missile.
...but rolling to prevent the impact of something that you've already rolled to avoid...I think that's taking two bites of the cherry and really just slows the game play down. I prefer to think of the outcome in two simple terms, either a person is hindered or they aren't...a situation is defused or it isn't...the specific details of the event can be narrated through the story.
He shoots me but I take no damage.
a) He missed
b) He hit, but my armour deflected the shot
c) He hit, but my armour deflected the worst of it, I suffered a flesh wound and now I'm frakkin' MAD!!! The adrenaline surging through my veins prevents me from suffering ill effects at the moment.
Each would be a different outcome according to dice with saving throws. Using said dice would take some of the narrative control out of the hands of the players, and deprive them of the chance of portraying their characters as they envision them. But then again, some people don't like to work their imagination overtime while playing, and they like following the whim of the dice.
Another reason I'm not a big fan of saving throws is that they are effectively story buffers in another sense. A GM can craft an elaborate trap, and a player can successfully make their save, effectively eliminating that part of the story. A player can be an absolute idiot through the course of a session, knowing that they have a saving throw that will probably negate the effects that other players can throw at them. With this in mind through, I guess that saving throws are a great tool for gamist play, they allow a player to step on up while reducing the chances of detrimental backlash.
Still, I'm not a big fan. Unless someone can really show me an elegant way of incorporating them into a story game.