I've been having this problem on my iMac (El Capitan) but not on my MBP (also El Capitan). Removing cookies, turning off AdBlock, disconnecting TimeMachine –– all suggestions I've found here and elsewhere –– seems to help... for about 30 minutes, and it's back to not opening "t.co" links again.
Specifically, Safari will spend a lot of time trying to open it and fail; after it fails and I reload, it will actually take me to the proper link.
Other shortened links (e.g., goo.gl) does not have this problem. Other browsers on my iMac (e.g., Chrome) does not have this problem. It's just Safari on my iMac, which is my main machine, and it's driving me crazy.