im not positive component cables can carry 1080i, if yours can then it may be that your tv only accepts 1080i through the hdmi port, as many do. cant say i have seen one that accepts 1080i through the component cables, i could be wrong though. i would try hdmi. save your receipt, worse case scenario you return it. upgrading to it is worth it, if that solves your problem.
Component cables should be able to use 1080i without issue, 1080P on the other hand could be a problem. Your TV may not auto detect the resolution changes though, you may need to set the TV into 1080i mode before the 360 will allow you to select that resolution. Also on the 360 make sure the picture format is in 16x9 and not 4x3.
In the TV settings is it set for 1080 ???? Some of them even though they are 1080 are not set that high .....for whatever reason I do not know. This happened with my fathers SONY when he got HD years ago. The picture looked like crap until I switched it in the menu to 1080. If the TV is set by default to 720 the xbox will not let you switch it to 1080.
yes component cables carry 1080i with a problem ( I did it for a long time ). As many of them were saying above it maybe something with your TV, it seems like your doing everything correctly with your xbox. Hope it works out for you.