Generally, yes, up to the TTL.
When using CDNs you usually configure TTLs (time to live) for your content. This is a maximum on how old the cache can get before it decides it absolutely has to refresh the cache with the newest content. For example, suppose you configure all *.jpg URLs to have a 5 minute TTL.
Then if your server goes down, you have an extra 5 minutes to bring it back up before users will notice. Well, at least for .jpgs. Well, at least for .jpgs that happened to have gotten cached beforehand.
Also, some CDNs use features like Akamai NetStorage where you can upload content directly to the CDN -- the CDN is given some content and told to serve it directly a priori. Since there's never an "on-demand" "pull" style caching going on here to begin with, that should of course work when your server is down.
As the other posters noted though, this isn't what CDNs are designed for and they provide NO guarantees that this behavior will work. It just happens to usually work (and it's awesome when you watch it happen!). And of course for specific technical details you'd have to contact your provider.