There are multiple ways to keep a backup or archive of your Twitter account. One of them is provided directly by Twitter. It's called the Twitter archive. Unfortunately, it has been very unreliable over the past weeks and sometimes doesn't even provide the download link for the archive within a week.
An alternative to the Twitter archive is using Birdfeeder. Once you sign up, Birdfeeder collects all your bookmarks and stores them. After that, you can use our powerful full-text search to find any Tweet you ever bookmarked instantly. Birdfeeder also provides a nice way to browse through your existing bookmarks.
If you are interested in receiving an archive of your Tweets as a CSV file, feel free to contact us. This is something that's currently in Beta as we are working on bringing it to all users.
Give it a try by signing up at https://getbirdfeeder.com to make sure you'll never lose access to your Twitter bookmarks.