I took a 737 test ride to check some of these ideas. OK it was MS Flight Simulator but it's probably a satisfactory model of the effects.
I could get it to 43000 feet with 100 fpm still possible. But it was very difficult. N1 was higher than perhaps is operationally realistic and EGT was over the red line a number of times.
At this height (in ISA presumably) we were in 'Coffin Corner' with the Barber's Pole at something like 225 kts - not much margin there for turbulence etc.
More significant was the lack of power above about 35000 feet.
I still can't quite get my head around why power falls away. The plane itself just goes faster as the air gets thinner until it thinks it's the same as at low level. Why does the engine run out of puff in the same circumstances?