Short answer: no.
Long answer: your system can only have one connection to any particular server for one particular thing open at a time. I.e., if you're downloading a file, or watching a movie. You can think of a connection as a plumber's snake (a spring you push through pipes to clear them) connecting your browser to the server. Even if you add a second pipe to your computer, the pipe can still only go through one of them, bandwidth isn't like water that can flow freely wherever there's space to go. You can, however, send it through the second pipe to another server.
In theory, it might let you get more total speed (i.e. download a file and watch a video at the same time). But only if your total internet connection is faster...