Spitzer Space Telescope infrared observations indicate that several transiting extrasolar giant planets have thermal inversions in their upper atmospheres. Above a relative minimum, the temperature appears to increase with altitude. Such an inversion probably requires a species at high altitude that absorbs a significant amount of incident optical/UV radiation. Some authors have suggested that the strong optical absorbers titanium oxide (TiO) and vanadium oxide (VO) could provide the needed additional opacity, but if regions of the atmosphere are cold enough for Ti and V to be sequestered into solids they might rain out and be severely depleted. With a model of the vertical distribution of a refractory species in gaseous and condensed form, we address the question of whether enough TiO (or VO) could survive aloft in an irradiated planet's atmosphere to produce a thermal inversion. We find that it is unlikely that VO could play a critical role in producing thermal inversions. Furthermore, we find that macroscopic mixing is essential to the TiO hypothesis; without macroscopic mixing, such a heavy species cannot persist in a planet's upper atmosphere. The amount of macroscopic mixing that is required depends on the size of condensed titanium-bearing particles that form in regions of an atmosphere that are too cold for gaseous TiO to exist. We parameterize the macroscopic mixing with the eddy diffusion coefficient K_zz and find, as a function of particle size a, the values that K_zz must assume on the highly irradiated planets HD 209458b, HD 149026b, TrES-4, and OGLE-TR-56b to loft enough titanium to the upper atmosphere for the TiO hypothesis to be correct. On these planets, we find that for TiO to be responsible for thermal inversions K_zz must be at least a few times 10⁷ cm² s^–1, even for a = 0.1 μm, and increases to nearly 10¹¹ cm² s^–1 for a = 10 μm. Such large values may be problematic for the TiO hypothesis, but are not impossible.