I know that a Noetherian local ring $R$ has finite global dimension if and only if $R$ is regular in which case $\mathrm{gldim}{R} = \dim{R}$. Therefore, for regular rings, every module has projective dimension at most $\dim{R}$. When $R$ is not regular there must exist modules with infinite projective dimension.
My question is:
For $R$ Noetherian local but not regular, do there exist modules with finite but arbitrarily large projective dimension or are the projective dimensions of modules with $\mathrm{pd}(M) < \infty$ bounded? If so, are they bounded by $\dim{R}$?
I'll assume all rings are commutative. In this case, your question was answered precisely by Auslander and Buchsbaum in "Homological dimension in local rings". Link!
The answer is yes: these finite projective dimensions are bounded by the Krull dimension of $R$!