I think a big problem with gaming (I refer mostly to RPGs) and their stories is because it's all so segmented. After all, the devs have to cater to both gameplay and story, often reinforcing the former to the detriment of the latter (which is especially bad when the former is tedious and dull). They cut the game in arbitrary sections, following obsolete gaming conventions that SHOULDN'T still be followed.
Cutscene/story mode, then exploration mode, talking to townies mode, dungeon mode, combat mode. Some of these, the way they're implemented NOW, hurt storytelling. When in dungeon mode, all odds are off! The protagonist turns into a vicious killer, murdering with impunity - with barely any consequences.
"So, you've wiped out this entire mansion? Do you realize what will happen now?"
"Whoops."
"Erm, well. I hope you have a good explanation."
"It was self-defense!"
"Okay, then we'll let you proceed to the next part of the story. This, of course, makes perfectly good sense story-wise."
It's things like these, where gameplay suppresses story's voice, that make the story suffer. But that's just one instance.