The hybrid finite-element and boundary-integral (FE-BI) method is a powerful numerical technique for computing scattering by inhomogeneous objects. Although the application of edge-based finite elements and the combined field integral equation (CFIE) in the FE-BI method has successfully removed the difficulties of the treatment of dielectric interfaces, sharp conducting edges and corners, spurious solutions, and interior resonance problems inherited in the original FE-BI method using node-based elements and the electric-field integral equation (EFIE) or magnetic field integral equation (MFIE), the FE-BI method still has a bottleneck which is the dense matrix generated by the boundary integral equation (BIE). Our renewed interest in the FE-BI method originated from the recent development of the fast multipole method (FMM) and the multilevel fast multipole algorithm (MLFMA). Our objective is to apply MLFMA to BIE to completely remove the aforementioned bottleneck for general 3D problems. During the course of pursuing this goal, we have encountered several problems associated with the efficiency and accuracy of the FE-BI method implemented using the edge-based elements and CFIE. This paper reports our study of these problems and the implementation of MLFMA in the FE-BI method.