You can watch hi def TV from coax, but not 1080p - 1080i is the max.
If the coax comes from an antenna you can watch all your local over-the-air stations in digital TV, if your TV has an ATSC tuner. Whether it's in HD is up to each station. There is no requirement that OTA be in HD. Some DTV broadcasts, esp. those on the new alternate channels, may be in 480p.
Even if the TV claims the signal is 1080i it may be coming from an upconverted standard-def source at the station. In which case you will see that source better than you ever saw it on analog, but it won't be high-def.
If the cable comes from cable TV, you can watch those same local over-the-air digital stations in whatever format they are sending, if your TV has a QAM tuner.
Max in either case is 1080i. Cable tv won't supply 1080p signals anyway, even via a set-top box. At present the only true 1080p source you can put in your home is a Blu-Ray player.
Note that you might be able to tell your set-top box to send 1080p, but this is another case of upconversion.. the source is no better than 1080i.
Either HDMI cables or component cables will get HD, all the way up to 1080p, from cable box or satellite box or Blu-Ray player to the TV. It is a myth that component cables will not carry HD.
Edit - added:
There is much confusion on this point. To get the local over the air (OTA) channels in HD from your cable without a cable box, you are going to have to tell the TV to search for the QAM channels. You'll recognize when you're on a QAM channel because the TV will display a "hybrid" channel number, like 8-1 instead of just 8.
You may also find alternate channels, like 8-2, carrying continuous newscasts, or weather, or alternate language programming, or something else... these will usually be at 480p. (And you may not be able to watch these without connecting the cable from the wall straight to the TV! The cable company doesn't have to carry them, and probably won't.)
If the TV is displaying just the regular OTA channel number you're watching the cable company's analog rendition of the OTA station, and no, it won't be any better than it ever was.
edit 2: No, the cable box does not have the e.g. 3-1 output from its RF.
However the raw cable does carry this. I'm talking about connecting the cable TV cable directly from the wall to the TV. If you do that you will be able to see the local OTA digital TV stations, in HD if they are broadcasting in HD.
It is an FCC requirement that all cable systems carry these (because it's an FCC requirement that they carry the local stations, and before long, the DTV signals are all there will be).
Now it's true that you won't see any OTHER HD or digital cable channels, like the premium channels, or even things like Universal HD. You have to go through the cable box to see these, and if you're going through the cable box, then you need either HDMI or component to get HD out of it.
And it is an outright LIE that component cannot carry HDTV. HDMI is NOT necessary. HDMI may have a very slight quality edge (most people can't see the difference), but it can also have problems, like not working because of poor HDCP implementations.