For SEO purposes, is there an optimal number of links within an article? Do too many links bring down your ranking in the search engines?

It all depends on the quality of the links.
is the link to similar content? Does the site backlink? PR? Sandbox?

It is too hard to find an optimal # but if you think it is too much, it probably is.

Agree with Quality of links
but if you have millions of quality links it will be excellent
and it can give you extra ordinary higher rankings
but if your link is from link farm page or poor quality class d links
then it will effect your search engine rankings

~ Hope that answer's the question ~

quality link is better I support it.

usually in article directories they require 2

As long as the links are quality you shouldn't have much trouble. Links pointing TO your article page is also an important factor as to how much value is available to give out with all those links. From what I know must article sites and paid blog posts usually allow up to 3 links. It also depends on the length of the article.

Article directories normally allow 2-3.