Have other Big Bangs happened before?

We don't know.

For a while, some cosmologists speculated that instead of one single Big Bang, there were many mini-Big Bangs long ago within the part of space that we now can study. The problem is that a universe created from many Big Bangs would be very lumpy looking. The cosmic background radiation is so smooth that only by some very specific fine tuning, could you get a lot of separate mini Big Bangs to end up producing something as phenomenally smooth as the background radiation.

Now, some theorists have speculated that our universe might be the spawning ground for other universes. At a scale near the Planck scale of 10^-33 centimeters, pieces of spacetime break off from our universe through wormholes. These pieces pinch off, and their wormhole connection to our spacetime evaporates, but the piece that has broken off then experiences its own Big Bang and inflates to become a separate universe which is completely disconnected from our own and totally unobservable. There is no way to test such a theory, so it is actually a non-scientific speculation.

So the bottom line is that we really don't know for sure, and we have no real good way to test this idea because we can't see beyond that part of space that came from the current Big Bang 13.7 billion years ago.