Here we report on a search, using Swift/XRT and Chandra, for X-ray emission from 28 SNe that belong to classes whose progenitors are suspected to be embedded in dense circumstellar matter (CSM). Our sample includes 19 Type IIn SNe, one Type Ibn SN, and eight hydrogen-poor superluminous SNe (SLSN-I such as SN 2005ap). Two SNe (SN 2006jc and SN 2010jl) have X-ray properties that are roughly consistent with the expectation for X-rays from a collisionless shock in optically thick CSM. However, the X-ray emission from SN 2006jc can also be explained as originating in an optically thin region. Thus, we propose that the optical light curve of SN 2010jl is powered by shock breakout in CSM. We suggest that two other events (SN 2010al and SN 2011ht) were too X-ray bright during the SN maximum optical light to be explained by the shock-breakout model. We conclude that the light curves of some, but not all, SNe IIn/Ibn are powered by shock breakout in CSM. For the rest of the SNe in our sample, including all of the SLSN-I events, our X-ray limits are not deep enough and were typically obtained too early (i.e., near the SN maximum light) for definitive conclusions about their nature. Late-time X-ray observations are required in order to further test whether these SNe are indeed embedded in dense CSM.