G. David, J.-L. Journé and S. Semmes have shown that if b_1 and b_2 are para-accretive functions on \mathbb R^n , then the « Tb Theorem» holds: A linear operator T with Calderón-Zygmund kernel is bounded on L^2 if and only if Tb_1 \in \mathrm BMO, T*b_2 \in \mathrm {BMO} and M_{b_2} TM_{b_1} has the weak boundedness property. Conversely they showed that when b_1 = b_2 = b , para-accretivity of b is necessary for the Tb Theorem to hold. In this paper we show that para-accretivity of both b_1 and b_2 is necessary for the Tb Theorem to hold in general. In addition, we give a characterization of para-accretivity in terms of the weak boundedness property and use this to give a sharp Tb Theorem for Besov and Triebel-Lizorkin spaces.