We investigate the possibility of constructing bicovariant differential calculi on quantum groups SO_q(N) and Sp_q(N) as a quantization of an underlying bicovariant bracket. We show that, in contrast to the GL(N) and SL(N) cases, neither of the possible graded SO and Sp bicovariant brackets (associated with a quasitriangular r-matrices) obey the Jacobi identity when the differential forms are Lie algebra-valued. The absence of a classical Poisson structure gives an indication that differential algebras describing bicovariant differential calculi on quantum orthogonal and symplectic groups are not of Poincare-Birkhoff-Witt type.