That card should be capable to do that. I would suspect that whatever cable you're using to go from composite to S-video is not feeding the proper vsync to S-video. Thats usually on the Green channel of RGB. If your display supports S-video output and you hooked up the the breakout cable properly, then you probably haven't switched the card input to s-video. If you have done this properly then all I can suggest is to locate another breakout cable and try again. I've seen cables which have bad connections in them while brand new. It happens rarely, but does happen. Did you install the capture utility provided with your card? Without that nothing works.
If you provide more detail about the exact setup you have, I can be more helpful. Its also possible that the Xbox has some sort of requirement in the video output loop thats not being met or other "environment" variable that is preventing a good signal (such as a raster loopback). Some signals may contain a "chain lock" which only allows one device attached at a time to the output channels. This prevents unsafe conditions that can possibly overheat the equipment and start fires. (Those are mostly older TV units from past decades that had no thermal protection built in that shuts it off in an overheat condition). I'm just speculating here without more information.
Your video chain should be Xbox/TV output/into the card. This requires a TV with an output. Most TV's are "input only" unless they specifically built in the feature on your TV. Most TVs made today do not have "outputs". Those are usually found on higher end equipment or dual purpose monitors/TVs. I use a Hauppauge card also but I don't record video games, however the card does have utilities to do that.
Note: Where possible, always use the same output as the input. Composite to composite. S-video to S-video, etc. This is because very few TV's can do live transcoding of video signal arrangement. In fact only broadcast quality monitors can do so that I've seen since 1992. There are cables which can convert the proper signal type & strength with composite to S-video but do so only as a function of the output (end) device or an input device with auto sense.