Actually there wasn't a lot of luck. Twitter is something that users actually wanted: a broadcast "status" update that wasn't just on their instant message client.
Twitter is successful because they remained positive and persevered while other people capable of making a similar system didn't do anything about it. Grumpy programmers just posted confused blog posts and forum comments about it. "Twitter? Huh? I don't get it." Nobody really tried to compete with them head on, and they still don't have any real competition. People say facebook is their main competition, and they are right, but Facebook has so much baggage of being something else that isn't Twitter that it seems unlikely people will ever consciously choose between one or the other. More likely people will use both services.
It's a bit of luck that anyone cares about posting their status online but it's not really luck that they are using Twitter to do so... What else would they use?