  Let $g$ be a finite dimensional simple Lie algebra. Denote by $\mathcal B$
the category of all bounded weight $g$-modules, i.e. those which are direct sum
of their weight spaces and have uniformly bounded weight multiplicities. A
result of Fernando shows that infinite-dimensional bounded weight modules exist
only for $g=sl(n)$ and $g=sp(2n)$. If $g=sp(2n)$ we show that $\mathcal B$ has
enough projectives if and only if $n>1$. In addition, the indecomposable
projective modules can be parameterized and described explicitly. All
indecomposable objects are described in terms of indecomposable representations
of a certain quiver with relations. This quiver is wild for $n>2$. For $n=2$ we
describe all indecomposables by relating the blocks of $\mathcal B$ to the
representations of the affine quiver $A_3^{(1)}$.
