It turns out the answer is "no". There is an example in section 3.4 of Riedtmann's paper "Degenerations for representations of quivers with relations", Ann. sci. Éc. Norm. Sup. v. 19 (1986), 275-301.
I won't reproduce the answer here (unless people complain), but one can see that it isn't a direct sum in the way that I asked for because in fact it is indecomposable.