Are there any atmospheric condition under which a visible raindrop can fail to hit the ground by evaporating first? I imagine this would require a large vertical temperature difference, and possibly the rain forming very high up. Has anything like this been observed experimentally, or if not - is it possible to perform a calculation to show whether this is plausible?
Yes, most certainly, and meteorologists call this kind of rain Virga (see Wikipedia page of the same name).
These are the salient and more interesting points of the Wiki article:
Often it is falling ice crystals that undergo compressional heating as the fall from greater heights, where the pressure is very low;
It is very common in desert and temperate climates: Western United States, Canadian Prairies, the Middle East and Australia.
It plays a role in seeding non-Virgal (i.e. reaching the ground as liquid) rain when virgal material is blown into another supersaturated cloud and begets rain through nucleation;
Its evaporation, with its high associated latent heat, means that virga draws a great deal of heat from the surrounding air, thus begetting violent up and downdraughts hazardous to aeroplanes;
Almost all (sulphuric acid) rain on Venus is virga. Presumable all rain on the early Earth was too.