The atmosphere does a pretty good job of slowing down the radiative escape of heat. But you can still freeze water in a desert overnight if you can thermally isolate it and expose it to the desert night sky.
Tyndall first computed that the Earth would be too cold to support life without its atmosphere and greenhouse gasses (Fourier also did the same computation). AIP has a nice page about his early work in the field:
It is known that ancient Egyptians knew how to make ice. Shallow porous clay dish of water on top of a straw bale exposed to clear night sky. eg.
The thermal inertia of the atmosphere can easily squash radiative cooling that is why frosts are common only on still nights.
They are not measuring it in the right way then. That number sounds about right for a cloudy night! The effective clear sky temperature is roughly speaking around 200K or -80C if you prefer.
It is only 4K in the microwave and radio bands where the atmosphere is almost completely transparent a window from 10MHz to 1THz with only the odd minor gap due to mostly due to water vapour.
Err. Sorry Phil no you don't. Power transmitted ~ T^4
T_sky^4 = (4^4 +0.5*280^4)/2 ~ 280/4^(1/4) = 200K
(about what is observed on average clear night)