The angular momentum only has quantized eigenvalues; this statement is valid quite generally for all bodies. For example, $J_z$ has to be a multiple of $\hbar/2$ because
$$ U = \exp(4\pi i J_z) $$
is the rotation by $4\pi$ and such a rotation brings every state to itself and has to be identity. (For a $2\pi$ rotation, the state changes the sign if it contains an odd number of fermions.) Therefore, we have
$$\exp(4\pi i j_z) = 1\quad \Rightarrow\quad j_z\in\{0,\frac 12, 1, \frac 32, \dots\}$$
Can the quantization of $j_z$ be actually measured? Well, one may only measure a sharp value of $j_z$ if the object is an eigenstate. Eigenstates of $j_z$ are rotationally symmetric with respect to rotations around the $z$-axis, up to an overall phase. So if we have a non-axially-symmetric object, its sharp $j_z$ eigenvalue obviously can't be observed because it's a linear superposition of many states with different $j_z$ eigenvalues.
For atoms, the angular momentum may be observed; these are the usual quantum numbers associated with the electrons. In the same way, the total angular momentum may obviously be measured and shown to be quantized for nuclei.
Larger systems are molecules. For some molecules, the quantized nature of the angular momentum may be measured. To add some terminology, we measure the rotational quantum numbers of these molecules by observing transitions in the rotational spectrum and the method is the rotational spectroscopy:
http://en.wikipedia.org/wiki/Rotational_spectroscopy
It only applies to molecules in gases because in solids and liquids, collisions constantly distort the angular momentum. Also, one can't have a well-defined quantized $j_z$ for "true solids" i.e. crystals because crystals aren't symmetric under continuous rotations; they're only kept invariant by the discrete crystalline subgroup of the rotational group.
So the maximum size for which the quantization may be verified are "rather large" molecules of gases and the maximum size is getting larger as the progress goes on (and as people are able to reduce the temperature and improve the accuracy).