They can be used for putting URLs in print materials such as newspapers - they are much easier to type by hand than long URLs. The email shortener thing adds light protection against bots harvesting emails. The Bitcoin address shortening feature lets you point to BTC addresses on fields where only URLs are allowed, and displays a useful QR code alongside the address. tny.im can also act as a pastebin, in case you haven't noticed (paste the text in the "Paste or type item to shorten here"). URL shorteners are also useful on IRC.

Personally, I use the link list feature to save a snapshot of open tabs when I'm reasearching something on a public computer and want to open the same set of pages later, at home or on another public terminal where I don't have access to conventional browser bookmarks. Some of my software's main download links are also tny.im links. I can edit the link to point to a new location if I need to change where the software is hosted. This way, the latest version is always reachable at the same address.

Actually, Twitter is probably a very bad reason to use link shortening these days, unless you are looking for click tracking (t.co doesn't do it), of course.