Rolo para sapatos is a micro-game that does this

The basic mechanic of the game is that you start off with the skill Do Anything 1 and if you want to do anything you have to roll a d6 to see if you succeed. If you roll a 6 you advance a rank in whatever you were trying to do. So if you were trying to break down a door and rolled a 6 you might get Door Breaker 2.

As you continue to do things in the game, anytime you roll all 6s on a skill role you gain a more specific, higher ranked habilidade.

While this game is mathematically simple, it is incredibly flexible and definitely genre agnostic. Realism is up to the narrator and characters.

Call of Cthulhu does this.

(I'm using 6th Ed. p53 for reference.)

When you successfully use a skill, you can mark it on your sheet (only one mark per skill). Then when the keeper (GM) calls for experience checks, you make a d% roll for each skill you've got a mark next to.

For each skill, if you roll higher than your current score in that skill (skills go from 001 to 100), you increase the score by 1d10. Then clear all the skill marks on the sheet.

This means the only way you can advance in a skill is if you've successfully used it since the last time the keeper called for experience checks.

It's a capricious system which creates a vicious curve: as you increase in skill, you're less and less likely to roll higher than your rank in it on an experience check, so higher skills advance much more slowly than lower ones. Even when you do succeed on the roll, you can get anywhere from 1 to 10 more ranks. However, any skill that isn't used regularly will remain unadvanced for sure, and that sounds like what you're aiming for.

Roleplaying básico

Chaosium's venerable BRP has percentile skills that advance in this fashion. When skills are used during play, they are marked. At the appropriate time (session end, if memory serves), the player rolls against the existing skill, hoping to falhar. If they fail, they roll again to see how much the skill has increased. Rolling to falhar means that it becomes harder and harder to advance as your skill increases - modeling many skills in reality. BRP has been used for many genres for many years.

Roda ardente

Luke Crane's modern classic also advances skills based on their use. You can see some more details aqui - but the basic idea is that you need a certain distribution of tests of varying degrees of difficulty against a skill in order to advance it. Again, the system is designed to make early increase easier than later. Burning Wheel is itself a fantasy system, but it has been adapted into games that cover several other genres, so I think it satisfies the genre-agnostic requirement.

Unknown Ponies: Failure is Awesome

While the game was designed to be played as a game set in the My Little Pony: A amizade é mágica setting, there's little tying the setting and system together. The system is also based off of Exércitos desconhecidos, which has a muito different setting from Unknown Ponies.

In Unknown Ponies, anything can theoretically be a skill¹. Unlike its parent game, where you spend experience points to increase skills, in Unknown Ponies your skill increases by 1 when you falhar a skill roll. (Hence, "failure is awesome.")

Because of this, the more you use a skill, the more unlikely it becomes for you to increase it further each time you use it². The game is a percentile roll-under system, so your stat is literally a percentage chance to succeed, with occasional situational modifiers (and increased for your "obsession skill" where you can always swap the roll's digits³, and whenever you utilize your Rage/Fear/Noble passions where conditionally once per session per passion you can swap the roll's digits or reroll).

One advantage here is that you don't have to keep track of twice as many numbers on your character sheet; there's no "experience bar" per-skill, you simply increase the skill when you fail. On the other hand, skill increases are dependent on variance, so some characters may have wildly higher skills than others. On the de outros other hand, if you have a "power gap" among your characters, the characters with lower skills are the ones that have actually been succeeding at their tasks. (Consider what would happen if skill increases were dependent on sucesso: the players that succeed would be more likely to succeed in the future, and the power gap would expand.)

This system also works well with the "yes, but..." style of storytelling. When a player attempts an action, they succeed⁴ -- the roll merely determines how bem they succeed, or what consequences may result from that action. In a percentile roll-under system where characters start with stats in the 25-35 range, they're going to fail rolls, frequently. When failed rolos don't mean failed ações, players tend to not become as frustrated with the system.

Attributes vs. Skills

This pattern does não apply to a character's atributos in Unknown Ponies. Attribute progression rules are more closely tied to the setting, and is basically spending experience points, like the parent game.

When a character in-game (or a player out-of-game) does something that exemplifies one of the five primary Elements of Harmony (Generosity, Honesty, Laughter, Kindness, Loyalty), that character (or the character owned by that player) receives a point on the associated Friendship Track. These points may be spent to manipulate the actions of NPCs, or alternatively you may spend five points in any combination to receive a point of the sixth Element, Magic (from the lore, the Element of Magic will appear when the other five are gathered). Magic points may be spent to manipulate the result of a die roll (turn critical failure into failure, failure into success, or success into critical success), activate skills the GM rules to be particularly powerful (for those familiar with the setting: think Dash's Sonic Rainboom, or Pinkie's... Pinkieness), or to increase attributes.

Of course, if you use the system and don't want the sunshine-and-rainbows part of it, you could simply fall back to using experience points⁵. Depending on the tone of the game you're running, you could also potentially disallow attribute increases entirely, leaving characters at their starting values. (Skills may not exceed their associated attribute.) Using Unknown Ponies character creation rules, starting attributes will be in the 30-60 range, or 30-80 for adult characters.

notas:

1. In my current campaign, one of my players (who has a habit of Nice Job Beaking It, Hero [warning: TVTropes]) joked about picking up "Apocalypse" as a skill. If he continues on the way he has, I may end up handing it to him. In one of the Exércitos desconhecidos published one-shots, one character has a skill "Making Gobs of Money," another has "Thuggery" and "Shooting Things," and a third has "Faking a Seizure." Standard character creation in Unknown Ponies gives 9 specific skills at 25, and either 2 racial skills at 25 or 50 skill points, depending on race selection.
2. I won't say "more difficult," as it all ultimately comes down to luck, and you may have some player that keeps rolling 00.
3. Unknown Ponies is designed to have players play as "blank flanks," children who have not yet discovered their Special Talent, and as such do not assign their Obsession Skill until they reach 40 in a skill. However, the game has rules for starting play as an adult, and Exércitos desconhecidos starts all characters with their Obsession Skill assigned.
4. In a sense, the combat system reflects this, too. Any "piercing" weapon, such as a dagger, will always deal at least 1 damage, even on a miss. Note that "yes, but..." is not explicitly a part of either Unknown Ponies or Exércitos desconhecidos; it's merely a style of GMing that can extend across games and genres.
5. In my campaign, I've generally been handing out 3-6 points of the primary 5 Elements per session per character on average, and 5 of those points may be turned into a Magic point, which can increase an attribute by 1. You can extrapolate from that how many experience points to dole out (and experience costs) if you want to fall back to that.

As a novel example, the original Battletech board game by FASA used to implement this in a statistical/probabilistic way. Everytime you rolled a double six on skill check, you would be able to advance your relevant skill. Which meant your skills advanced once around every 36 uses.

Universe did this -- you assigned a skill point to that skill whenever you rolled and got a 0. (and/or a 1 or 2, depending on the skill). After certain number of skill points, your skill level increased. The "0" meant that sometimes you received a point when you succeeded, sometimes when you failed.

http://en.wikipedia.org/wiki/Universe_(role-playing_game)

In practice, what this meant was my players would do silly things to force rolls to increase their skill. "I shoot the tree." While realistic, it didn't lead to a fun game.