The formation of interstellar water has been commonly accepted to occur on the surfaces of icy dust grains in dark molecular clouds at low temperatures (10-20 K), involving hydrogenation reactions of oxygen allotropes. As a result of the large abundances of molecular hydrogen and atomic oxygen in these regions, the reaction H2 + O has been proposed to contribute significantly to the formation of water as well. However, gas phase experiments and calculations, as well as solid-phase experimental work contradict this hypothesis. Here, we use precisely executed temperature programmed desorption (TPD) experiments in an ultra-high vacuum setup combined with kinetic Monte Carlo simulations to establish an upper limit of the water production starting from H2 and O. These reactants are brought together in a matrix of CO2 in a series of (control) experiments at different temperatures and with different isotopological compositions. The amount of water detected with the quadrupole mass spectrometer upon TPD is found to originate mainly from contamination in the chamber itself. However, if water is produced in small quantities on the surface through H2 + O, this can only be explained by a combined classical and tunneled reaction mechanism. An absolutely conservative upper limit for the reaction rate is derived with a microscopic kinetic Monte Carlo model that converts the upper limit into a maximal possible reaction rate. Incorporating this rate into simulations run for astrochemically relevant parameters, shows that the upper limit to the contribution of the reaction H2 + O in OH, and hence water formation, is 11% in dense interstellar clouds. Our combined experimental and theoretical results indicate however, that this contribution is likely to be much lower.