We address a number of puzzles relating to the proposed formulae for the
degeneracies of dyons in orbifold compactifications of the heterotic string to
four dimensions with $N =4$ supersymmetry. The partition function for these
dyons is given in terms of Siegel modular forms associated with genus-two
Riemann surfaces. We point out a subtlety in demonstrating S-duality invariance
of the resulting degeneracies and give a prescription that makes the invariance
manifest. We show, using M-theory lift of string webs, that the genus-two
contribution captures the degeneracy only if a specific irreducibility
criterion is satisfied by the charges. Otherwise, in general there can be
additional contributions from higher genus Riemann surfaces. We analyze the
negative discriminant states predicted by the formula. We show that even though
there are no big black holes in supergravity corresponding to these states,
there are multi-centered particle-like configurations with subleading entropy
in agreement with the microscopic prediction and our prescription for S-duality
invariance. The existence of the states is moduli dependent and we exhibit the
curves of marginal stability and comment on its relation to S-duality
invariance.