Normally the more links you have on your web page, the more Google and other search engines will love you and try to boost your search engine rankings. Part of most search engine algorithms has been to raise the ranking of a site that has a great deal of link popularity. This is why it is always essential to monitor the people who are linking to you. If their links are broken they will be damaging your site. Google, the grand master of this ranking methodology, often penalizes sites that provide links to links that serve no purpose or that is broken. This leads to a potentially very dangerous situation because you have no real control over what sites choose to link to yours unless your website is a blog in which case you can banish links from being attached to your site.
The drag is that the more popular your site is the more links you are going to have and the more time you are going to be spending checking to make sure that the links are operative.

There is a quick way to see if sites that are linked to your site are active. On Google download the Google toolbar and do a search on the site. If the site's link is broken it will not be green in color on a sliding green scale that is pictured on the toolbar. If they are gray get rid of them, as these sites are just dead weight in terms of links and probably already affecting your searchability on the Internet.