I am asking this question because I am currently working with the function $f$ defined by $f(x) = 1/(1+x^2)$ and interpolating it at $n+1$ for the interval $[-2,2]$. I have found that it theoretically is less.
My finding are as follow: \begin{array}{|c|c|c|} \hline \ \text{Value of $n$} & \text{Maximum error (approximation)} & \text{Maximum error (Theoretical)} \\ \hline 2 & 0.305573 & 0.299488 \\ \hline 4 & 0.1618 & 0.0950007 \\ \hline 8 & 0.0992211 & 0.0167722 \\ \hline 18& 0.0717574 & 0.000890209 \\ \hline \end{array}
Could there be cases where Lagrange bound is not satisfied?
It is important to mention the exact location of the interpolation nodes. The interpolation error using nodes $x_0, \cdots, x_n \in [-2,2]$ is given by $$ e_n(x) = \dfrac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=0}^n(x-x_i), $$
So the error can be bounded by
$$ \frac{\|f^{(n+1)}\|_{\infty} \| \prod_{i=0}^n(x-x_i)\|_{\infty}}{(n+1)!}. $$
Depending on the location of the nodes, the term $ \| \prod_{i=0}^n(x-x_i)\|_{\infty}$ can behave very differently (you should look into Runge's example and the use of Chebyshev nodes).