If you want to download Tweet for your account, you can request Twitter to download Twitter archive. Besides, you can use Twitter advanced search techniques to search tweets for a user by applying specific filter in three sections : word section, account section and date section.
But the technique can be helpful but not enough to get the exact data what you want. I will suggest you to use a Twitter service provider tool to get relevant Tweets data. You can go for Trackmyhashtag to find Tweets easily for any date range since 2006. Visit the website and visit the page "Old Tweets". On this page, you will find a form which you have to fill with your search term for which you need Tweets. After a while, you will get your Tweets data and other details with it.
I hope I helped!