If my understanding is correct, this is true of sufficiently nice nonabelian Lie groups (see Ben Webster's answer below), and any finite group. On the other hand, this is false for any infinite compact abelian group by Pontryagin duality (see Kevin Buzzard's comment below), and by extension for any group with such a group as a quotient (perhaps even as a quotient of a finite-index subgroup; see Ben Webster's comment below).

So: are there any nice conditions weaker than being a Lie group which guarantee that a compact group only has finitely many irreducible representations of each dimension? (Nice necessary conditions would also be interesting.)