Node embedding methods map network nodes to low dimensional vectors that can be subsequently used in a variety of downstream prediction tasks. The popularity of these methods has significantly increased in recent years, yet, their robustness to perturbations of the input data is still poorly understood. In this paper, we assess the empirical robustness of node embedding models to random and adversarial poisoning attacks. Our systematic evaluation covers representative embedding methods based on Skip-Gram, matrix factorization, and deep neural networks. We compare edge addition, deletion and rewiring strategies computed using network properties as well as node labels. We also investigate the effect of label homophily and heterophily on robustness. We report qualitative results via embedding visualization and quantitative results in terms of downstream node classification and network reconstruction performances. We found that node classification suffers from higher performance degradation as opposed to network reconstruction, and that degree-based and label-based attacks are on average the most damaging.