I just created a document library and changed permissions for it. I
edited the permissions for "contributors" so they could only "view"
items and not edit them. Now Readers and Contributors on the site
should have the same limited permissions in this particular library,
right?

Wrong. I then logged in as a contributor and tried to edit a document
in the library. I was able to do this with no problems.

Help! I need to make sure that only web designers and above have the
ability to edit these documents.