Thanks, Kevin. A URL is always worth more than a thousand words.
When you really want to see what is happening, look a the page's source in your browser.
In this case, here is what the "eabutton1" banner looks like:
<a href="/DesktopModules/Admin/Banners/BannerClickThrough.aspx?BannerId=67&VendorId=16&PortalId=0"
class="NormalBold" target="_new"
rel="nofollow">
<u>eabutton1</u>
</a>
Here you can see that the "eabutton1" text is not in the alt tag, but is the actual text content of the link that is generated.
If you look at the banner template, you'll probably be able to spot what is wrong.