If $R$ is commutative of dimension d, then we have:

The Bass Cancellation Theorem: If $M$ has rank $\ge d+1$ after localizing at any prime, and if there exists a finitely generated projective module $Q$ such that $Q\oplus M\approx Q\oplus N$, then $M\approx N$. (You probably don't even need to assume $N$ projective for this, as long as you know that $M$ is.)

Plumstead's Theorem: If $R=A[t]$ for some noetherian ring $A$, then you can replace $d+1$ with $d$.

In the non-noetherian case you can work with j-dimension instead of dimension (the j-dimension of $R$ is the length of a maximal chain of ideals, each of which is an intersection of maximal ideals).

If $R$ is noetherian of dimension d, then we have:

The Bass Cancellation Theorem: If $M$ has rank $\ge d+1$ after localizing at any prime, and if there exists a finitely generated projective module $Q$ such that $Q\oplus M\approx Q\oplus N$, then $M\approx N$. (You probably don't even need to assume $N$ projective for this, as long as you know that $M$ is.)

Plumstead's Theorem: If $R=A[t]$ for some noetherian ring $A$, then you can replace $d+1$ with $d$.

In the non-noetherian case you can work with j-dimension instead of dimension (the j-dimension of $R$ is the length of a maximal chain of ideals, each of which is an intersection of maximal ideals).