My simulation below shows the expected 20V output with a 20Ω load.
(Note that R7 should be 120Ω for the LM317 to maintain regulation with no load.)

Looks like a questionable LM317 model.
Try mine below.

View attachment 194792

Thanks! That's much better and the output voltage is correct too as in the other one it was a little higher.

 

In this case, when the current increases such that the first LM317 reduces the voltage to the second LM317 (U1) below about 6.7V, U1 drops out of regulation, reducing the output voltage and thus limiting the current.

That was the key point at least for me to understand, you were very helpful again. That the following LM317 drops out and limiting happens. I did a lot of simulations and on the breadboard (up to 250mW combinations so far) and works like a charm. It works so well that I am in dilemma, to have them connected permanently or not. When lowering the current resistor of the first one to 0Ω it behaved like almost it wasn't there. But I like switches... and disconnecting things when they are not necessary. Most of the times it is the voltage regulation that will be in need only. Using a multi throw switch I would isolate the first LM317.

 

I am in dilemma, to have them connected permanently or not.

The advantage of leaving it out, is that you eliminate its 2V or so drop, which then allows the output voltage to be adjusted to a higher voltage.

Using a multi throw switch I would isolate the first LM317.

It can be keep out of the circuit with a simple SPST switch that just shorts it out by connecting its input to the second LM317's input.