We report on the coronal hole (CH) influence on the 54 magnetic cloud (MC) and non-MC associated coronal mass ejections (CMEs) selected for studies during the Coordinated Data Analysis Workshops (CDAWs) focusing on the question if all CMEs are flux ropes. All selected CMEs originated from source regions located between longitudes 15E - 15W. Xie, Gopalswamy, and St. Cyr (2013, Solar Phys., doi:10.1007/s11207-012-0209-0) found that these MC and non-MC associated CMEs are on average deflected towards and away from the Sun-Earth line, respectively. We used a CH influence parameter (CHIP) that depends on the CH area, average magnetic field strength, and distance from the CME source region to describe the influence of all on-disk CHs on the erupting CME. We found that for CHIP values larger than 2.6 G the MC and non-MC events separate into two distinct groups where MCs (non-MCs) are deflected towards (away) from the disk center. Division into two groups was also observed when the distance to the nearest CH was less than 3.2×105 km. At CHIP values less than 2.6 G or at distances of the nearest CH larger than 3.2×105 km the deflection distributions of the MC and non-MCs started to overlap, indicating diminishing CH influence. These results give support to the idea that all CMEs are flux ropes, but those observed to be non-MCs at 1 AU could be deflected away from the Sun-Earth line by nearby CHs, making their flux rope structure unobservable at 1 AU.