Maybe technically true as a matter of biology in isolation, but mostly not in practice — specifically due to that "something they ate or drank" part. The basal body temperature reduction that happens at night really isn't relevant for anyone who consumes any amount of caffeine during the day (which is most people), as when caffeinated, the BBT shift downward at night isn't accompanied by a complementary upward shift in
comfortable/tolerable air temperature. Rather the opposite.
Caffeine, as a cholinergic, constricts peripheral blood vessels, and reduces bloodflow; and as a diuretic, it also reduces interstitial-fluid retention in peripheral tissues. These effects combine to decrease your skin temperature (or rather, to make your skin temperature less reflective of your core temperature and more reflective of the ambient air temperature); while very slightly increasing your core temperature (and blood pressure! Which is one reason caffeine is bad for your heart!)
Most of your heat-sensing nerves are in your periphery, not in your core. So caffeine, by making your skin cooler, makes you feel cooler (even though your basal body temperature goes up!) to which your body responds by sweating less (even though caffeine, as a cholinergic, would force you to sweat in great-enough amounts. See: SLUDGE syndrome.)
Most people don't drink caffeine after a certain time in the evening; and so at night, whatever caffeine was in their bodies has a chance to flush out — which then suddenly allows bloodflow, blood pressure, and interstitial fluid to wash back out to their skin and extremities — and with that comes an increase in skin temperature, "hot" qualia, and triggered sweating from signalled overheating (i.e. the same reason you sweat from spicy food even though your BBT isn't increasing.)