I want to clarify your subject line. The body of your post is correct; black bars on the left and right DO mean that you're not watching HD (at least, in the vast majority of cases). However, your subject line is not true as a general statement, because you will almost always have black bars when watching modern movies. The difference is that the bars will be on the top and bottom, instead of the left and right (though there are times when you might get some of both). For those who don't understand this, here's the reasoning.

Television networks do a pretty good job of standardizing aspect ratio to 16:9. They do this because they know most of their viewers have 16:9 TVs. As such, the vast majority of modern television programs will not show black bars when watched on a 16:9 widescreen TV.

HOWEVER:

Movies are a different story. Movies are produced in a variety of aspect ratios these days, and most DVD/Blu-ray copies attempt to faithfully reproduce the original source. Unfortunately for owners of typical 16:9 ( 16/9 = 1.78 ) widescreen TVs, most current movies are being produced in either 1.85, or 2.39 (if I remember right). In both of these cases, you will see horizontal bars. In the case of 1.85 movies, the bars are small. In the case of 2.39, they're fairly significant.

So in conclusion, "if the picture isn't filling the screen, it's not HD" only applies if you're watching TV. If you're watching a movie on a 16:9 TV, you're probably going to have some sort of letterboxing no matter what; this doesn't mean you're not watching HD.