We present the results from an Hubble Space Telescope/ACS search for supernovae associated with X-ray flashes 020903, 040701, 040812 and 040916. We find strong evidence that XRF 020903 (z=0.25) was associated with a SN 1998bw-like supernova and confirm this using optical spectroscopy at t ~ 25 days. We find no evidence, however, for SN 1998bw-like supernovae associated with the other three events. In the case of XRF 040701 (z=0.21), we rule out even a faint supernova similar to SN 2002ap, using template light-curves for several local Type Ic supernovae. For the two cases in which the redshift is not known, XRFs 040812 and 040916, we derive robust redshift limits assuming they were accompanied by supernovae similar to SN 1998bw and compare these limits with photometric redshift constraints provided by their host galaxies. We supplement this analysis with results for three additional events (XRFs 011030, 020427 and 030723) and discuss the observed diversity of supernovae associated with X-ray flashes and gamma-ray bursts. We conclude that XRF-SNe exist, but can be significantly fainter than SN 1998bw, possibly consistent with the observed spread in local Type Ibc supernovae.
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