For the most part, humans have good sight. A lot of time and effort during our modern era is put into making TV and computer screens at a higher and higher resolution in order to make things look as "real" as possible - that is, to make the pixels onscreen indistinguishable from what we would normally see. But how good are our eyes really? Lets find out.
Before all of this, I'll direct you to a nice, short, but informative link (https://xkcd.com/1080/), courtesy of xkcd. A good representation of how we see, it outlines the many different parts of vision very nicely. Focusing primarily, however, on the "resolution" of our field of vision, that is, how many "pixels" we can see, we can see it varies. Right in the center few degrees, in the foveal region, we can see stuff quite clearly, which makes sense, because we're looking at it. However, the blurred characteristic of the surrounding areas isn't just because we aren't focused on it, but because there simply isn't as much data provided there - much less, in fact. While our center of vision is comparable to a high-res camera, the surrounding areas are much worse quality, with the entire area outside of the center ~10 degrees containing a fraction of the data that the center area does. Our brain just fills in the gaps.
So while at times our vision is quite good, other parts could use some work. And even with the high detail of our foveal region, you still might not need that new HDTV. Based on how far away you're sitting, it might not even be noticeable, so don't waste your money.