Additive synthesis can be more than just stacking a bunch of sine waves to emulate an instrument. For instance are the sine waves following the harmonic series or some other series of harmonics? are the sine waves at a constant amplitude? are the sine waves in the same phase?
The key thing to remember is that almost all natural sounds will have dynamic events that are in constant change so only stacking waves in a static manner will not render a complex tone.
One might start with making an analysis of what tone or sound you are interested in replicating. Here is the 101 on this:
TAKING THE WAVEFORM APART
THE MATHEMATICS OF ELECTRONIC MUSIC
Frequency modulation synthesis
and a tutorial:
This is not trivial work.