Well, no, never. Games from the point of view of ability to solicite emotional responses are still pretty weak next to films/books. Films in particular have the advantage of requiring minimal imaginition as they present apparently real people in emotional situations.
I can say though that Silent Hill 2 and SOTC made me feel genuinely sad and moved for the characters, and ICO made me smile broadly at the end of the credits.
But for the most part while I might feel a mild emotional response it's nothing like watching a film or reading a book.
Maybe one day. I guess looking at Heavy Rain and The Last Guardian there are creative souls trying.
With The Last Guardian in particularl if your pet 'whatever' howls horribly when hurt by arrows and the like that might be a toughie as I'm a sucker for animals and that thing looks very, very cute just from the trailers never mind after bonding with it during the game.
Try to be reasonable... its easier than you think...