We study the stability of multiple conducting edge states in a topological insulator against all multiparticle perturbations allowed by time-reversal symmetry. We model a system as a multichannel Luttinger liquid, where the number of channels equals the number of Kramers doublets at the edge. We show that in a clean system with N Kramers doublets there always exist relevant perturbations (either of a superconducting or charge density wave character) which always open N-1 gaps. In the charge density wave regime, N-1 edge states get localized. The single remaining gapless mode describes the sliding of a "Wigner-crystal"-like structure. Disorder introduces multiparticle backscattering processes. While single-particle backscattering turns out to be irrelevant, the two-particle process may localize this gapless mode. Our main result is that an interacting system with N Kramers doublets at the edge may be either a trivial insulator or a topological insulator for N=1 or 2, depending on the density-density repulsion parameters, whereas any higher number N>2 of doublets gets fully localized by disorder pinning, irrespective of the parity issue.