One of our newer clients recently had the misfortune of discovering that their Twitter name was not available. The ideal short name for this non-profit seems to belong to someone in Somalia who posted two posts back in April of 2007 and otherwise lays dormant. A frustrating state of affairs akin to finding that your preferred website domain name is already occupied.
When it comes to Twitter names, you can of course use variants of your organization's name, taking on a "Canada" or simply a "CA" to the end of your Twitter handle or using some other variant of your name. The thing to be aware of there however, is that when limited to posting 140 characters a too long Twitter handle can actually get in the way, particularly when others want to retweet your post and therefore include your @TwitterName in their post. (Also note that Twitter currently restricts names to 15 characters).
If your organization is not yet convinced about using Twitter, you may want to make a case for grabbing your name (or a reasonable variant of it), if it's still available. Even if you don't get to using it for a few months, you'll want to secure it.
So what can you do?
Currently Twitter's policy is to only release names that are registered by someone else if it is a trademark violation (see here for their word on this). They are talking about releasing accounts that have been inactive for at least 18 months at some point, but do not currently have a time frame in place for when they will do this. They, sadly, will not do it on an individual request basis either. (The official inactive user names policy can be found here).
TweetClaims offers a free service that will notify you when your name of choice becomes available (or a pro service for $20/year which will actually text message you as soon as your name becomes available).