The main result of the paper is that the Lifshits-Krein trace formula cannot be generalized to the case of functions of noncommuting self-adjoint operators. To prove this, we show that, for pairs (A₁, B₁) and (A₂, B₂) of bounded self-adjoint operators with trace class differences A₂-A₁ and B₂-B₁, it is impossible to estimate the modulus of the trace of the difference f (A₂, B₂) - f (A₁, B₁) in terms of the norm of f in the Lipschitz class.