In Hardy & Wright "An Introduction to the Theory of Numbers" there are two theorems:
Theorem 233: There are positive rationals which are not sums of two non-negative rational cubes.
Theorem 234: Any positive rational is the sum of three positive rational cubes.
The first one is proven by providing a counterexample - the number $3 \in \mathbb Q$, the second one is constructively proven using elementary number theory.
Now I wondered, can we classify the rationals $r \in \mathbb Q$ that satisfy theorem $233$ - the rationals that are not sums of one or two (but three) positive cubes?