The issue is most likely that the icons/images in the css files are using relative paths, so if your bundle doesn't live in the same app relative path as your unbundled css files, they become broken links.
We have rebasing urls in css on our todo list, but for now, the easist thing to do is to have your bundle path look like the css directory so the relative urls just work, i.e:
new StyleBundle("~/Static/Css/bootstrap/bundle")
Update: We have added support for this in the 1.1beta1 release, so to automatically rewrite the image urls, you can add a new ItemTransform which does this rebasing automatically.
bundles.Add(new StyleBundle("~/bundles/publiccss").Include(
"~/Static/Css/bootstrap/bootstrap.css",
"~/Static/Css/bootstrap/bootstrap-padding-top.css",
"~/Static/Css/bootstrap/bootstrap-responsive.css",
"~/Static/Css/bootstrap/docs.css", new CssRewriteUrlTransform()));
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