To bring up a classic of good game writing (which may have already been brought up, but bears repeating): Chrono Trigger. A complex plot spanning millenia, in which you must move back and forth in time to gather the power, the resources, and the people you'll need to save the world from the monster that lives in its core, and in spite of all this, the story made sense. On top of that, the dialogue was pretty good, the characters were interesting (Magus, willing to be seriously evil in order to save the world; Frog, tormented by the memory of an old life he can't get back but driven by the memory of the hero he once was; Robo, the machine who gains a soul through acts of self-sacrifice; even the silent protagonist Chrono is well fleshed out, and the rest aren't so bad either), the plot twists were always a nice surprise, and the game accounted nicely for the paradoxes that time travel necessarily creates.
I also liked Majora's Mask, but that one didn't do time travel quite as well as Chrono Trigger. Not surprising, given the constraints, but it raised the issue of paradoxes and didn't properly resolve it. Still, it was cool to discover the world you were in, the mess it was in, how it got that way, and how to stop it. The scene where you catch the falling moon was a beautiful moment of triumph, and the ending left you wondering about the nature of evil spirits and why the act the way they do.
Now, on to the subject of bad writing: I say that good writing trumps no writing, but no writing trumps bad writing. If the game doesn't need a story, don't give it one. People have mentioned DOOM and Pac-Man and BeJeweled. Certain games need no exploration of motives any more than chess does. You just charge in and start playing.
On the other hand, if it does need a story, give it a good one. If you need a story to move it along, the story ought to be one of the best parts of the game. If you must tell the player why he's doing what he's doing, you'd better make sure that the player's reaction is not laughter.
Now, on the subject of cutscenes: I'm generally against them. There are good ones, but I'm of the opinion that in games, interactivity should never stop. Why show people doing cool stuff when you could be letting them play cool stuff?