We consider the existence and spectral stability of multi-pulse solitary wave solutions to a nonlinear Schrödinger equation which incorporates both fourth and second-order dispersion terms. We do this for both the bright and the dark soliton regimes. We first show that a discrete family of multi-pulse solutions exists, which is characterized by the distances between consecutive copies of the the primary solitary wave. In the bright soliton regime, we then reduce the spectral stability problem to computing the determinant of a matrix which is, to leading order, block diagonal. Under additional assumptions, which can be verified numerically and are sufficient to prove orbital stability of the primary solitary wave, we show that all bright multi-solitons are spectrally unstable. By contrast, using a similar approach in the dark soliton regime, we find that dark multi-solitons can be spectrally neutrally stable. Finally, we show results of numerical spectral computations, which are in good agreement with our analytical results. This is supplemented with numerical timestepping experiments, which are interpreted using our spectral computations.