For the incompressible Navier-Stokes equations, classical results state that weak solutions are unique in the so-called Ladyzhenskaya-Prodi-Serrin regime. A scaling analysis suggests that classical uniqueness results are sharp, but current nonuniqueness constructions are far below the critical threshold. In this talk, we show the nonuniqueness of weak solutions near a borderline of the Ladyzhenskaya-Prodi-Serrin regime. In particular, we construct a class of non-Leray-Hopf weak solutions that are smooth outside a set of arbitrarily small Hausdorff dimension in time. Joint work with Alexey Cheskidov.