Then a miracle happens.
The dense cloud cover starts to dissipate under the sunlight's warmth, splitting apart to reveal a stark rocky peak way above us.
Set within it is the cable car station, an improbable construction wedged into the rock, like the lair of a James Bond villain.
As we reach it, I suddenly remember that there’s nothing beneath our feet for a very long way; and then we gently ease into place, 2634 metres above sea level.
Ascending stairs, we find ourselves in an unexpectedly classy cafe, though most of us continue through onto a viewing platform in front of the building, partly built into the rock and jutting out into thin air.
Eventually I go inside, wondering what drink is most suitable to celebrate such an experience (Whisky? Beer?), and end up with a frosted glass of demanovka, a traditional Slovak liqueur.
Back outside, there’s an odd mood of exhilaration in the air.
I think, like me, everyone is thinking how improbable this all is, that human beings shouldn’t be this high up from the earth, and certainly not sipping alcoholic beverages while doing so.
We’re all delighted at somehow being part of this impossible thing, and braced by the strange mix of material comforts and an underlying sense of danger.
But such atypical periods of one’s life - removed from the concerns of the mortals below us - have to come to an end.
After 50 minutes our return car arrives and we make our descent.
I have lunch at the restaurant at Skalnaté Pleso, enjoying a decent Hungarian goulash.
Then it’s out onto the rocks for a hike along the mountainside west to Hrebienok, from where a funicular railway leads back down to Starý Smokovec, terminating just above the hotel.
After all this physical effort, the sight of the Grand is a relief, and its wellness centre a godsend.
Wrapped in a sheet, I take a well-earned dose of relaxation, moving between the sauna, the steam room, the infrared sauna with its weird colour-changing globe, the ice-cold pool and the heated ceramic beds in the tepidarium.
Then, on my way back to my room, I bump into Joan and Joan, the two women I’d noticed in the cafe earlier, and join them for tea.
They’ve been going off every day on tours around the countryside; tomorrow they’re rafting down the Dunajec River on the border with Poland.
Since we’re getting along like a house on fire, I mention the hotel’s resemblance to a Christie plot item, and they agree.
Daringly, I suggest they’d fit right in to the story, and they laugh. “Like Miss Marple?” says one.
Still on a high from my ascent of the mountain, I can only smile. The only mystery surrounding the attractions of the Tatra Mountains, is why they aren’t better known outside Slovakia.