Is it consistent with ZFC to have a cardinal $\kappa$ which is not Ramsey and $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$ holds for some $n\in \omega$?

The partition relation $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$ holds iff for every function $f:[\kappa]^{<\omega}\rightarrow \omega$, there is an $A\in [\kappa]^\kappa$ such that for each $l\in \omega$, $|f''[A]^l|\leq n$.

Ramsey cardinals have $\kappa \rightarrow (\kappa)^{<\omega}_\omega$, which is equivalent to $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,1}$.

It seems to be similar to a "$n$-Rowbottom cardinal", but any definition I've seen of $\gamma$-Rowbottom cardinals are for $\gamma>\omega$, which makes me wonder if "$n$-Rowbottom" is just equivalent to Ramsey.

For example the definition from Kanamori's book:

"If $\omega < \gamma < \kappa$, then $\kappa $ is $\gamma$-Rowbottom iff $\kappa \rightarrow [\kappa]^{<\omega}_{\lambda,<\gamma}$ for any $\lambda<\kappa$"

I guess I'm asking if $n>1$ and $\kappa \rightarrow [\kappa]^{<\omega}_{\omega,n}$ implies $\kappa \rightarrow (\kappa)^{<\omega}_\omega$. I don't think it does, but it would be nice to see an example where cardinals with partition relations of this form are used.