There's different category of writting and good storytelling.
In the end, a game is more about the experience than the story, so the writing needs to be good but the overal story-telling need not be "incredible".
However, you can get to games like Mass Effect which just make you straight up cringe all the time and the only concepts to understand from every line is thrown in your face in the most obnoxious ways.
I think its just incredibly obvious how huge the difference is when the writters obviously tried to reach out some semblance of intelligence, and when games like Mass Effect just try to reach out to teenagers.
What's an example of good writing? Sorry if that's a dumb question; I'm not much of story buff.
It's not common at all in video games... I'd say Portal 1/2, and... older adventure games like The Secret of Monkey Island, Full Throttle...
Planescape: Torment is another commonly cited example of good writing in a video game. I don't have that much experience with them personally, but I've also seen things like Grim Fandango and the various Matsuno games (Final Fantasy Tactics, Vagrant Story) brought up.