The answer is yes! And it has been used in this way many times. To pick a random, recent example by Vogt et al. (2014) - a 4-planet system orbiting a K0V star.
The problem is akin to separating out multiple frequencies in a blended chord and is accomplished with fourier analysis techniques.
A useful public resource ("Systemic") containing some software that can do this task. It's great and there are a number of helpful tutorials on how this can be used to find multiple planets in radial velocity data.
Your statement about the motion of the star is not correct. The position of the star with respect to the centre of mass of the entire system will vary in a complex way (e.g. think about a star plus planet; the centre of mass is on a line joining the two. But now introduce another planet and the centre of mass will move off this original line, but its position will move depending on the relative position of the second planet to the first.)
Here is a useful picture I show students. It is the position of the solar system centre of mass with respect to the Sun's position. If the solar system only contained Jupiter, this track would be roughly a circle executed at the orbital period of Jupiter around the Sun. This is indeed the dominant factor, but the other planets contribute significantly.