We investigate many-body localization in isotropic Heisenberg spin chains with the local exchange parameters being subject to quenched disorder. Such systems incorporate a nonabelian symmetry in their Hamiltonian by invariance under global SU(2)-rotations. Nonabelian symmetries are predicted to hinder the emergence of a many-body localized phase even in the presence of strong disorder. We report on numerical studies using exact diagonalization for chains of common spin length 1/2 and 1. The averaged consecutive-gap ratios display a transition compatible with a crossover from an ergodic phase at small disorder strength to an incompletely localized phase at stronger disorder. Studying the sample-to-sample variance of the averaged consecutive-gap ratio, we distinguish this incompletely localized phase from the fully many-body localized phase by its scaling behavior.