We report on exploratory Chandra observations of five galactic nuclei that were found to be X-ray bright during the ROSAT all-sky survey (with L_X > 10^43 erg s^-1) but subsequently exhibited a dramatic decline in X-ray luminosity. Very little is known about the post-outburst X-ray properties of these enigmatic sources. In all five cases Chandra detects an X-ray source positionally coincident with the nucleus of the host galaxy. The spectrum of the brightest source (IC 3599) appears consistent with a steep power-law (Gamma~3.6). The other sources have too few counts to extract individual, well-determined spectra, but their X-ray spectra appear flatter (Gamma~2) on average. The Chandra fluxes are ~10^2-10^3 fainter than was observed during the outburst (up to 12 years previously). That all post-outburst X-ray observations showed similarly low X-ray luminosities is consistent with these sources having `switched to a persistent low-luminosity state. Unfortunately the relative dearth of long-term monitoring and other data mean that the physical mechanism responsible for this spectacular behaviour is still highly unconstrained.