  We prove the ill-posedness of Leray solutions to the Cauchy problem for the
ipodissipative Navier--Stokes equations, when the dissipative term is a
fractional Laplacian $(-\Delta)^\alpha$ with exponent $\alpha < \frac{1}{5}$.
The proof follows the ''convex integration methods'' introduced by the second
author and L\'aszl\'o Sz\'ekelyhidi Jr. for the incomprresible Euler equations.
The methods yield indeed some conclusions even for exponents in the range
$[\frac{1}{5}, \frac{1}{2}[$.
