The power of several neutrality tests to reject a simple bottleneck model is examined in a coalescent framework. Several tests are considered including some relying on the frequency spectrum of mutations and some reflecting the linkage disequilibrium structure of the data. We evaluate the effect of the age and of the strength of the bottleneck, and their interaction. We contrast two qualitatively different bottleneck effects depending on their strength. In genealogical terms, during severe bottlenecks, all lineages coalesce leading to a star-like gene genealogy of the sample. Some time after the bottleneck, once new mutations have arisen, they tend to show an excess of rare variants and a slight excess of haplotypes. On the contrary, more moderate bottlenecks allow several lineages to survive the demographic crash, leading to a balanced genealogy with long internal branches. Soon after the event, data tend to show an excess of intermediate frequency variants and a deficit of haplotypes. We show that for moderate sequencing efforts, severe bottlenecks can be detected only after an intermediate time period has allowed for mutations to occur, preferably by frequency spectrum statistics. Moderate bottlenecks can be more easily detected for more recent events, especially using haplotype statistics. Finally, for a single locus, the bottleneck results closely approximate those of a simple hitchhiking model. The main difference concerns the frequency distribution of mutations and haplotypes after moderate perturbations. Hitchhiking increases the number of rare ancestral mutations and leads to a more predominant major haplotype class. Thus, despite a number of common features between the two processes, hitchhiking cannot be strictly modeled by bottlenecks.