  Let $\mathscr{A}$ be an abelian category having enough projective and
injective objects, and let $\mathscr{T}$ be an additive subcategory of
$\mathscr{A}$ closed under direct summands. A known assertion is that in a
short exact sequence in $\mathscr{A}$, the $\mathscr{T}$-projective
(respectively, $\mathscr{T}$-injective) dimensions of any two terms can
sometimes induce an upper bound of that of the third term by using the same
comparison expressions. We show that if $\mathscr{T}$ contains all projective
(respectively, injective) objects of $\mathscr{A}$, then the above assertion
holds true if and only if $\mathscr{T}$ is resolving (respectively,
coresolving). As applications, we get that a left and right Noetherian ring $R$
is $n$-Gorenstein if and only if the Gorenstein projective (respectively,
injective, flat) dimension of any left $R$-module is at most $n$. In addition,
in several cases, for a subcategory $\mathscr{C}$ of $\mathscr{T}$, we show
that the finitistic $\mathscr{C}$-projective and $\mathscr{T}$-projective
dimensions of $\mathscr{A}$ are identical.
