Yes and maybe.
With a software encoding card, you can generally turn off the encoding steps (and any extra filtering the card's software does), and just watch the raw feed. With most hardware encoding cards (like the Hauppauge PVR line), your video card can't access the video frames until they've gone through the encoder chip. While this only takes a fraction of a second, it's enough to make playing any sort of twitch game (like a fighting game) nearly impossible, or at least very annoying. Something like an RPG would not be a problem.
One solution (without getting another card) would be to split the video/audio feed, so it runs to both a TV (which you would use to play) and the capture card (which you would use to record).