We study Shafarevich-Tate groups of motives attached to modular forms
on Gamma0(N) of weight bigger than 2. We deduce a criterion for
the existence of nontrivial elements of these Shafarevich-Tate groups,
and give 16 examples in which the Beilinson-Bloch conjecture implies
the existence of such elements. We also use modular symbols and
observations about Tamagawa numbers to compute nontrivial conjectural
lower bounds on the orders of the Shafarevich-Tate groups of modular
motives of low level and weight at most 12. Our methods build upon
Mazur's idea of visibility, but in the context of motives instead of
abelian varieties.