If you go to the wikipedia page about numerical aperture, you will see that the definition has nothing to do with lenses, but with the half-angle of the light cone ($\theta$, assuming everything in air):
$NA=sin(\theta)$
From here it is "easy" to understand that for a single lens one can approximate it as:
$NA\approx\frac{D}{2f}$
With $D$ the diameter of the lens and $f$ its focal length.
However, as soon as you put two lenses in a row, the definition has to be done for the light cone, rather than for a single element.
I will however follow up with the simplest case, using approximations, to show that the NA can be much higher. Imagine two lenses with the same diameter ($D_1=D_2=D$), same focal length ($f_1=f_2=f$), separated by 1/4 of their original focal length. We can use the effective focal length formula to find the new $f_n$ effective focal length of such a system:
$\frac{1}{f_n}=\frac{1}{f_1}+\frac{1}{f_2}-\frac{d}{f_1f_2}$
with $d$ as the separation between the two lenses. Then we get:
$f_n=f/1.75$
which in turn, using the approximation $NA\approx\frac{D}{2f}$ means that: $NA_n=1.75*NA_o$, where $NA_n$ is the new effective NA of your system when compared to the original of a single lens $NA_o$. Not only is the new NA much higher, the second lens can also be down to $D_2=\frac{3}{4}D$. As long as $D_2>\frac{3}{4}D$ there is no influence whatsoever on the resulting NA. Being 3/4 smaller means that its NA is also 3/4 of the original.
As an exercise, draw the system I just described and you will see that the half-angle is much steeper for the new system.
So, in conclusion, yes, the NA is defined from the half-angle and it has nothing to do with the single lens NA of a compound system, which can in the end be effectively much higher.
