I just finished the Walking Dead game by Telltale and emotionally it made me feel more than most movies I've watched recently, especially the ending. I teared up completely. It's story is engaging, and the characters are believable and so well voice acted that it doesn't matter that it was a game.
If you haven't tried it I'd really recommend it.
So I say yes! More games are needed that emphasise story/plot and well rounded characters over graphics. Heavy Rain was another good attempt to try and do this.
PM me and I'll add you to my 3DS - feel free to add my NNID for WiiU too!