There is a famous topological result:
Let $X$ be a smooth manifold of dimension $n$, $E$ be a vector bundle of rank $k > n$, then $E$ contains a trivial line bundle.
So, I guess that (enlightened by Hartshorne's Exercise 2.8.2):
Let $A$ be a ring (some more conditions are needed, say Noetherian, universally catenary,...) of Krull dimension $n$ and let $P$ be a projective module of rank $k > n$. Then there exists a split injection $A \hookrightarrow P$
I wonder if there's a result similar to this one.