The 100m figure is for an active network — switches or hubs that are powered and repeating an amplified signal from one network node to another. Your router and XBox are nodes, and apparently they don’t have a strong enough signal to talk at 50 feet. Either that, or the cable itself is bad.
I’d try the 50ft cable between a PC and the router. If that works, then the offender is the XBox, since obviously the router is pulling its weight. If the PC doesn’t work, then it could be any of the three — the cable, the router, or the XBox. The only way to know would be to try a different 50ft cable, or put a quality switch in place — shorter cables from router to switch and from switch to XBox. Even a $20 switch should work fine: http://shop3.outpost.com/product/4774329
Good advice. I’ll try it.
Thank you.