schemes - Can one check formal smoothness using only one-variable Artin rings?

Let $f:Xrightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{prime}left[tright]/t^{n}$, where $k^{prime}$ is also a field?

Considering cuspidal curves one can show that you do at least need arbitrarily large $n$.