We consider the Schrodinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that according to the size of the initial data, a superposition principle may or may not hold. Surprisingly, it holds for large data, like in the linear case, but not for smaller ones. The proof relies on WKB analysis: for large data, we use the theory known in the case of a single initial WKB state, and properties of compressible Euler equations, while for smaller data, nonlinear interactions are present at leading order.