I enjoy playing a RPG that lets people be human. Back in the day playing D&D or Palladium's "Teenage Mutant Ninja Turtles"/"Robotech"/"Rifts" you were not often playing a true person but more like a video game character from "Double Dragon". You had no fear, there was no reason to hang back.. as long as you had hit points and your sword ( gun/mecha/etc) you were all set.
The first time I played GURPS it occurred to me that playing a game could be more "real". Now, not real in the tedious way and also not real in "skill roll for everything, including tying shoes", but real in that the character might get scared or that a person's motivations in playing a game are more than just achieving the next level. I had old skool D&D players tell me how much the levels and hit points made sense, but really it is a stand-in for the ideas that later made video games possible (levels and hit points are the basis of most video games). I can't see my own life where I am seeking experience points to get to my next level, I am not seeking a better understanding of Cisco Routers so that when a guy punches me I can take it better. The old systems were so one dimensional.
*major confession time*
I am too old to understand Twitter and I have never played D&D 3E, 3.5E or 4E.
I know you all think less of me, but I will continue writing this...
Many games need a sense of tension and fun that only real life "S*&T Happens" type stuff can offer. If your 4th level Cleric could always "Inflict Critical Wounds" then you are just a video game cabinet away from Gauntlet. If your Cleric was allergic to cats and was -4 dexterity in a sneezing fit around them, now things are interesting.
In other games such things were forced. In Rifts if you faced a Daemon there might be some forced "save vs insanity" roll, where you hit up a table on failure and ended up with a fear of heights. But I think when a character confronts a life or death situation for the first time (ever or even "in a while") and the blood gets pumping, there should be a roll. Anything a rookie "Glitter boy" does the first time they get face to face with "Azemodeous the eater of souls" is the right thing to do (even if they have to change their pants afterward).
In my game, such a reaction roll would be base upon the Mental Endurace/Will attribute. There are two kinds of rolls, contest and non-contest... This would be a non-contest roll. In a non-contest roll you must roll under half the attribute plus any applicable skill and modifiers. A normal person has a ME/Will of 10, with no skill (+0) and default modifier (+/-0) would have to roll under a 5 with a 20 sided die -- or 25% to hold it together. Now let's say that they Glitter Boy Rookie had seen video of Azemodeous dancing with a puppy on YouTube, he might get a +3 for a modifier which would have him roll under a 8 with a 20 sided die, or about 40% chance to keep it together.
The same goes with Tunkar the Warrior. If Thunkar is out in the woods and stops to look at a butterfly, a sneaky DM might ask Thunkar to make a Wisdom roll.
"Why?" says Thunkar's player.
"I will tell you later." says the DM.
Thunkar has a Mental Power/Wisdom of 8. Half 8 is 4, with no modifiers or skills applied by the sneaky DM Thunkar has a 20% chance and completely fails the roll.
Later, as Thunkar is ambushed by two Lollipop Trolls, he reaches for his two-handed axe and is surprised to find out he left it next to the pretty butterfly.
I am sure this stuff is possible with every system out there. I am sure you have even played D&D with such things happening. If you have not then I encourage you to find a way to add it to your games. It is a good way to prevent Munchkin-ism ( "Can I drop my charisma to 2 and raise my strength to 18?" "Yes -evil laugh- I am sure charisma will not come in handy.").