It is true that one of the component inputs shares a composite input. Are you getting any kind of error message when attempting to hook up the cable box via HDMI?

It could be that the cable box is sending out too low of a resolution. So even though the cable box might reset, you want to make sure that it is outputting a 720p 1080i/p signal. By going into the menu of the cable box, you can set resolutions. Once you do that, you should be able to get a signal from the cable box to the TV.

It's pretty rare that HDMI ports stop working, but it's also entirely possible. But in most cases, it's because either the cable box is set to send via component (and the customer doesn't switch the output to HDMI), or because the resolution is 480i (which is often the default in the cable box).

Keep me posted.