it is hard to describe, but it looks like just little bits of static around objects and words.

Are you getting the picture from a data compressed source? For instance, a DVD, a Tivo, a digital satellite, or a television station that's rebroadcasting data compressed material? If so, you might possibly be seeing ordinary MPEG artifacts. It depends on what you mean by static. I wouldn't know without seeing an example. I'm sure there's sites on the web that have magnified pictures of this sort of thing, anyone got a link?

Is this a common problem with big screen televisions?

It's common for people to suddenly notice MPEG artifacts that they hadn't seen before, yes. Because the picture is suddenly so much bigger. Lots of people got big screen TVs and suddenly realized that digital satellite TV was crap because they data-compress things too much. DVDs are better but still not perfect.

If you know what MPEG artifacts look like and you're sure that isn't what you're seeing, then no, it's not normal for a TV of any size to do that.

Edit: Something funky happened, either with the BBS messing up posts or with a post getting edited while I was replying. I no longer see the post I was replying to. In any case, my answers still stand.