It very much depends. Although many HDTV tuners and cable boxes have high definition component outs, up-sampling DVD players can only output a (upsampled) 720p/1080i picture over DVI-HDCP. It is likely that newer electronics, such as HDVD/Bluray players will not be equipped with Component outs-- only HDMI.
So, if you're shopping for a new display, you'll probably want HDMI, or DVI inputs.
Well, actually there are a number of DVD players that will upscale over component. Not many but there are some. The downside to this is that most if not all of these players will NOT upscale copy protected material over the component connection.
HD is very much accessible over the component connections. Any new display that you consider should include HDMI/DVI inputs. Some have one or the other. I have seen Sony and Panasonic displays that have both.
This answer is a broad one, that I assume should be asked more specifically for the precisely the reasons from the other answering posts. In real-world, it depends.
Theoretically, component video absolutely can handle HD signals and beyond.
But as noted many devices don't take full advantage of their analog component video outputs. So you should probably ask more specifically. But YPbPr as a transmission method and format is capable of HD no problem.