Abstract Many barred galaxies, possibly including the Milky Way, have cusps in their centers. There is a widespread belief, however, that the usual bar instability, which occurs in bulgeless galaxy models, is impossible for cuspy models because of the presence of the inner Lindblad resonance for any pattern speed. At the same time, there is numerical evidence that the bar instability can form a bar. We analyze this discrepancy by performing accurate and diverse N-body simulations and calculating the normal modes. We show that bar formation in cuspy galaxies can be explained by taking into account the disk thickness. The exponential growth time is moderate (about 250 Myr) for typical current disk masses, but it increases considerably (by a factor of two or more) if the live halo and bulge are substituted by a rigid halo/bulge potential; the pattern speeds remain almost the same. Normal mode analysis with different disk mass favors a young bar hypothesis, according to which the bar instability has saturated only recently.