An operator on a separable Hubert space is called bitriangular if it and its adjoint have upper triangular representations with respect to two (perhaps different) orthonormal bases. Although bitriangular operators have some tractable properties and seem to be the right context for generalization of matrices to infinite dimensions, we give counterexamples to various open problems regarding this class of operators. The counterexamples make use of a property that an M-basis may or may not have.