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.
Well then, it's time for a 3rd-party Twitter search engine, using a P2P-based scraper, with no regard for API rules or ToS. In the long term, what's more important: Twitter's bottom line, or content not being thrown down the memory hole?
Yes, but isn't it worth capturing that content, and in doing so, their users? The search engine I am suggesting could easily transition to being the next Twitter. We grab the content, their users perform searches, and then the users migrate when they see that they can not only search every tweet ever made (since we began indexing) but also add to the database directly (kind of like Google having both a search box and an "add your own URL" box on the main page).
I'd guess that the vast majority of twitter users don't know anything about this API shenanigans, blocking 3rd party clients and all this bullshit. They don't know and they don't care.
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.