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Excess near-infrared emission is detected around one fifth of main-sequence stars, but its nature is a mystery. These excesses are interpreted as thermal emission from populations of small, hot dust very close to their stars (`hot exozodis'), but such grains should rapidly sublimate or be blown out of the system. To date, no model has fully explained this phenomenon. One mechanism commonly suggested in the literature is cometary supply, where star-grazing comets deposit dust close to the star, replenishing losses from grain sublimation and blowout. However, we show that this mechanism alone is very unlikely to be responsible for hot exozodis. We model the trajectory and size evolution of dust grains released by star-grazing comets, to establish the dust and comet properties required to reproduce hot-exozodi observations. We find that cometary supply alone can only reproduce observations if dust ejecta has an extremely steep size distribution upon release, and the dust-deposition rate is extraordinarily high. These requirements strongly contradict our current understanding of cometary dust and planetary systems. Cometary supply is therefore unlikely to be solely responsible for hot exozodis, so may need to be combined with some dust-trapping mechanism (such as gas or magnetic trapping) if it is to reproduce observations., 18 pages, 9 figures, accepted for publication in MNRAS