Most lossless compression (like the algorithms used in gzip, bzip2, and zip) works by eliminating long repeated series of bytes in a file. As a bit of a contrived example, let's say your file has several instances of 100 spaces: a compressed version of the file might create a very short code that means 100 spaces, and replace those instances with this.
The catch is that for this to work, the file has to have a considerable number of repeated sequences of bytes. Most ordinary files do, which is why compression generally works well, but compressed files usually don't (that is, after all, the point of compression). That's why double-compression doesn't usually work very well: after you compress a file once, you've already taken out most of the things that made it compressible. Interestingly, it is possible to make files that compression would actually make larger, but from a realistic standpoint, files like that do not occur very often.
Lossy compression isn't so different, really. In its simplest sense, it alters the file in certain ways to make it compress better, but it tries to do this in ways that the user won't notice. In audio compression, these are called psychoacoustic algorithms, because it's all about changing the audio in ways that a human mind won't detect; I assume there's a similar word for video compression, but I don't know what it is. Anyway, once they've done these tricks to make the file more compressible, then they compress it in the usual way. But this means that lossy-compressed files also don't have many of the things that lossless algorithms look for.
Because of this, I'm afraid you're mostly out of luck. @WarrenYoung's answer can work if you have access to the original video, but if you only have these .mpg files, you'll need to be careful about re-encoding them: you won't break the file or anything, but the results might not look as good as if you had started with the original file. Different lossy algorithms throw out different kinds of data, but effects of doing that can stack up and become noticeable. When you're done re-encoding, you should watch the file to make sure it still looks good.