It is commonly assumed in deep survey work that all galaxies with X-ray luminosities above about 10⁴²ergs^-1 harbour AGN, regardless of their optical spectrum, due to the absence of starburst galaxies in the local Universe reaching such extreme luminosities. We investigate the validity of this assumption using new Chandra and XMM-Newton observations of three relatively nearby (z∼0.1) starburst galaxies, chosen on the basis of their SDSS optical spectrum, and their coincidence with a bright X-ray source from the 2XMMp catalogue. If originating from the galaxy in question, the X-ray luminosity is in each case greater than 10⁴²ergs^-1 (and in one case reaches 10⁴³ergs^-1), meaning these would be the most X-ray luminous starburst galaxies yet discovered if no AGN were present. However, the new X-ray data shows two of the three sources to display X-ray spectral and temporal variability between observations, with all three appearing point-like at Chandra resolution. Furthermore they are spectrally dissimilar to starburst X-ray emission, possessing power-law dominated spectra more reminiscent of AGN. We therefore conclude that otherwise innocuous AGN are indeed fuelling the X-ray emission from these galaxies, and that the assumption starburst galaxies do not exceed 10⁴²ergs^-1 remains observationally justified.