It has appeared recently that the underlying degree distribution of networks may play a crucial role concerning their robustness. Previous work insisted on the fact that power-law degree distributions induce high resilience to random failures but high sensitivity to attack strategies, while Poisson degree distributions are quite sensitive in both cases. Then much work has been done to extend these results. We aim here at studying in depth these results, their origin, and limitations. We review in detail previous contributions in a unified framework, and identify the approximations on which these results rely. We then present new results aimed at clarifying some important aspects. We also provide extensive rigorous experiments which help evaluate the relevance of the analytic results. We reach the conclusion that, even if the basic results are clearly important, they are in practice much less striking than generally thought. The differences between random failures and attacks are not so huge and can be explained with simple facts. Likewise, the differences in the behaviors induced by power-law and Poisson distributions are not as striking as often claimed.