For example, such a result was used 
by Ribet for GL_2 (1990) to prove that
the Taniyama-Weil Conjecture implies Fermat's theorem. 
In his thesis (1989), Taylor introduced 
a general method to raise the level for 
GL_2, to construct the global Langlands correspondance for 
classical Hilbert modular forms. Bellaiche (2002) 
generalized this for any rank one group, and used it to prove parts of the
Bloch-Kato conjecture. In a recent joint work, we obtained a general 
statement for U(3).