That happened to my desktop when I would play The Sims 2. I'd be able to play for a bit, then the screen went black and into stand by, and when I'd try to turn it on again, it would go straight into stand by again; I'd have to restart the computer to get it back on. My dad and I had deduced that either my graphics card was going bum, or my motherboard was burning out. Turns out, the fan on my graphics card was burned out and coming loose, which...Obviously isn't good. I'd say take it out, make sure nothing is physically wrong with the card itself, and of course check it in detail since there are so many tiny components. If it looks fine, if you have another computer, try switching the cards and play FFXI with the other one. If it works fine, it's your card. If not, you have bigger issues.

Hope that helps a little!