We describe experimental difficulties, and associated artefacts, that arose during the study of thin epitaxial films of the proposed multiferroic materials BiFeO3 and BiMnO3. The problems experienced include large leakage currents and charge injection from Nb-doped SrTiO3 substrates used as bottom electrodes. Charge injection prevents ferroelectric hysteresis loop closure and contributes to the apparent polarization. Leakage currents also contribute to the apparent polarization, and may help explain the large range of polarization values reported in the literature. Convincing reports of a material that is ferromagnetic and ferroelectric at room temperature remain elusive, and indeed we fail to reproduce a previous work concerning Tb-substituted BiFeO3. Our magnetization measurements show this material to be only a weak ferromagnet (0.1 mu(B)/unit cell). Lastly, we show that magnetoelectric measurements are easily corrupted in the presence of large leakage currents.