It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$.

Can I assume that, if one uses polynomials of degree $p$ and the respective function to be interpolated $f\in C^p([a,b])$, that the interpolation error of this spline is $O(h^{p+1})$? Is something like this present in literature?