Yes you can!
Step one: Download MyMedia - File Manager (blue logo with folder and blue dots on it) from Appstore
Step 2: Open MyMedia and at the bottom you will see "Browser" click on it. Type in
http://twittervideodownloader.com/
Step 3: Enter in Tweet URL. Example:
https://twitter.com/tjsullivan_/status/954467539257626629
Step 4: Click on Download, save.
Step 5: Go to "Media" tab right next to the browser tab, click on ur Twitter video and then click "Export to Camera Roll".
Not tried, but i believe you can also download YT videos via Clipconverter etc...
Enjoy