Bosonic two-dimensional self-bound clusters consisting of N atoms interacting through additive van der Waals potentials become unbound at a critical mass m*(N); m*(N) has been predicted to be independent of the size of the system. Furthermore, it has been predicted that the ground state energy E(N) of the N-atom system varies exponentially as the atomic mass approaches m*. This paper reports accurate numerical many-body calculations that allow these predictions to be tested. We confirm the

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... ence of a universal critical mass m*, and show that the near-threshold behavior can only be described properly if a previously neglected term is included. We comment on the universality of the energy ratio E(N+1)/E(N) near threshold.