This is a major misunderstanding of the Big Bang theory. The Big Bang theory isn't the beginning of the universe, but the beginning of the universe as we know it. i.e. when it started to expand!
Of course, if Big Bang theory is also the start of time, then it is quite possible for the universe to have "always existed", simply because there is no "always" without time.
Of course, if Big Bang theory is also the start of time, then it is quite possible for the universe to have "always existed", simply because there is no "always" without time.