Using XMM-Newton, we undertook a dedicated project to search for X-ray bright wind-wind collisions in 18 WR+OB systems. We complemented these observations with Swift and Chandra data sets, allowing for the study of two additional systems. We also improved the ephemerides, for these systems displaying photometric changes, using TESS, Kepler, and ASAS-SN data. Five systems displayed a very faint X-ray emission (log [L[SUB]X[/SUB]/L[SUB]BOL[/SUB]] < -8) and three a faint one (log [L[SUB]X[/SUB]/L[SUB]BOL[/SUB]] ∼ -7), incompatible with typical colliding wind emission: not all WR (Wolf-Rayet) binaries are thus X-ray bright. In a few other systems, X-rays from the O-star companion cannot be excluded as being the true source of X-rays (or a large contributor). In two additional cases, the emission appears faint but the observations were taken with the WR wind obscuring the line of sight, which could hide a colliding wind emission. Clear evidence of colliding winds was however found in the remaining six systems (WR 19, 21, 31, 97, 105, and 127). In WR 19, a large absorption and larger emission at periastron are even detected, in line with expectations of adiabatic collisions.