If the URL of the resource remains the same, the CDN servers (and the browsers) are free to cache them. So, if you are using CDN, you need to force a URL change every time the file content changes (commonly done by adding a version string).
Since, it is working for you, either your files are not getting served from the CDN at all or somehow the URL is getting updated.
Look at the URL from where your style sheet is getting fetched (network tab in the browser's debugger). Make sure the URL path is actually from the CDN and not your website directly.
If you have a MVC.net app and you are using System.Web.Optimization.BundleCollection for style bundle, it add a query parameter to the URL embedded in the HTML and changes it if the file contents change. This ensures that the stale cached copies of the resources are not used.
See CDN and bundle caching sections at http://www.asp.net/mvc/overview/performance/bundling-and-minification