Yes, and when APIs fail: P2P screenscraping. Regardless of its legality, as some other commenters here have brought up, Twitter can't stop distributed reading and storing without blocking lots of IP addresses.
Ha, twitter can't even block @mention spam from fresh accounts using very spammy words.
They have their target audience (people following celebrities) guaranteed for now. We're just a typical pain in the ass nerds.
As long as Justin Bieber and other celebrities continue to post, users will be there.
Also, the typical tweet doesn't make sense a couple of hours/days after being posted. I'm not sure if there's such interest in searching for old stuff.