Observations of carbon nanotubes under exposure to electron beam irradiation in standard transmission electron microscope (TEM) and scanning electron microscope (SEM) systems show that such treatment in some cases can cause severe damage of the nanotube structure, even at electron energies far below the approximate 100 keV threshold for knock-on damage displacing carbon atoms in the graphene structure. We find that the damage we observe in one TEM can be avoided by use of a cold finger. This and the morphology of the damage imply that water vapour, which is present as a background gas in many vacuum chambers, can damage the nanotube structure through electron beam-induced chemical reactions. Though, the dependence on the background gas makes these observations specific for the presently used systems, the results demonstrate the importance of careful assessment of the level of subtle structural damage that the individual electron microscope system can do to nanostructures during standard use.