Statistical analysis of satellite data shows a positive correlation between aerosol optical depth (AOD) and total cloud cover (TCC). Here we compare the slope of the linear regression between the logarithm of TCC and the logarithm of AOD, or the strength of the relationship, as derived from three satellite data sets to the ones simulated by a global aerosol-climate model. We analyze model results from two different simulations with and without a parameterization of aerosol indirect effects, and using dry compared to humidified AOD. We find that none of the hypotheses discussed in the literature is able to uniquely explain the positive relationship. The most important contribution in the model is from the swelling of aerosol in the vicinity of clouds, where relative humidity is high. The model also shows contribution of the aerosol cloud lifetime effect to the positive relationship, which, however, is of lesser importance.