Broken links are a real problem -- they are one of the most annoying parts of the web, and they waste a lot time. To give you one real world example, take a look at this Question of the Day. This is a question on a rather delicate topic, but it is one of the most frequently visited questions in the archive.
When that question was originally published, it had eight links associated with it at the bottom of the question. They were all interesting, relevant links. Within five months, all eight of the links were broken. Because all of the links were broken, it generated a lot of angry e-mail. New links were put in. They all broke, and so on.
HowStuffWorks contains many thousands of links. At regular intervals, we run a piece of code that checks for broken links. Each time, there are almost a thousand broken links.
Why does this happen? Here are the three most common reasons:

• Web site restructuring is by far the most common reason. For example, a couple of years ago, NASA decided to consolidate and move many of their sites. When they did, they reorganized their directory structure and changed just about every URL across NASA. As a result, every link that HowStuffWorks had to any page on NASA broke.
• People change ISPs. If you build a little personal Web site on AOL, and then change to another ISP and move your site in the process, anyone who linked to you is left with a broken link.
• Companies and universities have dynamic populations. When a student leaves a university, he/she loses the free Internet account on the university machines. All links pointing to that site break.

The bottom line is that anytime that someone moves a site or changes the names of the files in a site in any way, everyone linking to the site gains a broken link. If you have a Web site, you want to do your best to never change the file names or the directory structure (or, if you do, provide automatic redirecting from old names to new). The easiest way to do that is to register a unique URL for it and leave the file names the same for eternity. If you don't, you lose a lot of traffic!