Twitter is notorious for its disappearing tweets. Since Twitter is about sharing your most recent information, it only displays the most recent 3,200 tweets of an account. This does not mean that previous tweets are gone, they just can't be publicly viewed. Many overcome this by archiving their tweets.
Downloading Your Twitter Archive
Through your account settings, you can make a request to download your Twitter archive, which is every tweet you have made. It can take a few days to get the archive, but it is a convenient way to gather your history in one place.
Archiving Tweets with twDocs
One Twitter archiver is twDocs. Through this service you can save Twitter tweets, favorites, mentions, followers, friends, direct messages, and search results as PDFs, DOC, HTML, Excel, or other common file types. Best of all, you can save more than your own tweets! You can use this service for free, though there are export limitations.
Important: When using twDocs, don't save over your old archives, but rather make a new archive each time you create one. This will prevent you from losing old tweets you already pulled when they are no longer publicly visible. You may want to differentiate each one by including the date in the file name.