  Let $R$ be a commutative Noetherian ring with non-zero identity and $\fa$ an
ideal of $R$. Let $M$ be a finite $R$--module of of finite projective dimension
and $N$ an arbitrary finite $R$--module. We characterize the membership of the
generalized local cohomology modules $\lc^{i}_{\fa}(M,N)$ in certain Serre
subcategories of the category of modules from upper bounds. We define and study
the properties of a generalization of cohomological dimension of generalized
local cohomology modules. Let $\mathcal S$ be a Serre subcategory of the
category of $R$--modules and $n \geqslant \pd M$ be an integer such that
$\lc^{i}_{\fa}(M,N)$ belongs to $\mathcal S$ for all $i> n$. If $\fb$ is an
ideal of $R$ such that $\lc^{n}_{\fa}(M,N/{\fb}N)$ belongs to $\mathcal S$, It
is also shown that the module $\lc^{n}_{\fa}(M,N)/{\fb}\lc^{n}_{\fa}(M,N)$
belongs to $\mathcal S$.
