Summary: We prove the ill-posedness for the Leray-Hopf weak solutions of the incompressible and ipodissipative Navier-Stokes equations, when the power of the diffusive term \((-\Delta)^\gamma\) is \(\gamma <1/3\). We construct infinitely many solutions, starting from the same initial datum, which belong to \(C^{1/3-}_{x,t}\) and strictly dissipate their energy in small time intervals. The proof exploits the “convex integration scheme” introduced by C. De Lellis and L. Székelyhidi jun. [Ann. Math. (2) 170, No. 3, 1417–1436 (2009; Zbl 1350.35146)] for the incompressible Euler equations, joining these ideas with new stability estimates for a class of non-local advection-diffusion equations and a local (in time) well-posedness result for the fractional Navier-Stokes system. Moreover, we show the existence of dissipative Hölder continuous solutions of Euler equations that can be obtained as a vanishing viscosity limit of Leray-Hopf weak solutions of suitable fractional Navier-Stokes equations.