Let $U_q$ denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that $q$ is a complex root of unity of odd order and that $U_q$ is %the quantum group version obtained via Lusztig's $q$-divided powers construction. We prove that all regular projective (tilting) modules for $U_q$ are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for $U_q$. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type $B_2$ and in this case as well as for type $A_2$ we calculate explicitly the Loewy structure for all regular Weyl modules. We also demonstrate that these results carry over to the modular case when the highest weights in question are in the so-called Jantzen region. At the same time we show by examples that as soon as we leave this region non-rigid tilting modules do occur.