The 'wobble' is measured, and the resulting dataset is analysed to see what's there.  The measurement makes no assumption about what the results are.

The method can detect multiple planets around one star.  This is because different planets orbit at different distances from the star, and the period of time for one orbit is directly related to the distance from the star.  This allows different planets to be discerned from the 'wobble'-data.  The amount of wobble caused by each planet gives the planet's mass relative to the star.  The star's mass is known using standard astrophysical methods, so the mass of the planet can be inferred. 

Actually, the magnitude of 'wobble' depends on the planet's orbital period as well as it's mass.  This means the wobble-method preferentially detects large planets orbiting close to the star, since they have the biggest wobbles.  Small planets, and planets orbiting further out, have smaller wobbles, which are harder to detect because small changes in wobble are harder to measure accurately than big ones are.

So to answer your question about other planets accentuating or cancelling the wobble from one planet - yes, they can do this, but in such a way that the period of the other planets can be worked out and removed from the data, so they have no effect on the results (except that the researchers have discovered extra planets).