In case you hadn't heard, back in August of last year Twitter changed the rules for their API, limiting developers to 100,000 individual user tokens for outside apps (or 200% of then-totals, if the app already had more than 100,000 users). To say the change was controversial would be an understatement. Falcon Pro, a favorite among Android Twitter users, has hit the limit. New users cannot log into Twitter via Falcon Pro. That includes those who have already paid for the app and are reinstalling for a new device or ROM.
Falcon Pro was, and technically still is, one of the most popular paid Twitter clients on the platform. According to the Falcon Pro Twitter account, only 40,000 users have actually paid for the app, with the remainder presumably made up by pirated users. Apps in excess of 100,000 user tokens or 200% of their original users require additional permissions from Twitter to exceed the limit - so far, Twitter has not deemed any third-party clients on any platform worthy, with the possible exception of first-party clients and acquisitions like TweetDeck.
The developer has started a petition to extend Falcon Pro's token limit, but its success would depend upon Twitter's generosity. Since they haven't changed the policy in the last six months, it's unlikely that even an impassioned cry from users will do much good. It's generally believed that Twitter's new token policy is specifically designed to steer users towards the web interface and Twitter's own apps, which some see as a betrayal of the client culture that helped the service thrive in the first place. While other, older apps like Plume seem to be surviving, it's only a matter of time before growing pools of Android and Twitter users stretch their limits as well. The API token issue has already affected popular new clients on other platforms, like Tweetro for Windows 8.
The separate Falcon widget uses a different set of tokens, and appears to be operational. If you previously used Falcon Pro and switched to another app, you can release your personal token and open it up for one more user by going into your Twitter settings and removing Falcon Pro from your Apps list.
Update: Commenters have noted that users who have stopped using Falcon Pro, or returned it within Google Play's 15-minute refund window, may be artificially inflating the number of active user tokens.