The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether it is drifting to, or oscillating. Whenever the Lévy process drifts to, we prove that the th moment of the first passage time (conditioned to be finite) exists if and only if the th moment of the Lévy jump measure exists. This generalizes a result shown earlier by Delbaen for Cramér-Lundberg risk processes. Whenever the Lévy process drifts to, we prove that all moments of the first passage time exist, while for an oscillating Lévy process we derive conditions for non-existence of the moments, and in particular we show that no integer moments exist.