The result that you mention in the first part of your question is a classical result by Faber

> G. Faber, Uber die interpolatorsche Darstellung stetiger Funktionen, Jahresber. der deutschen Math. Verein. 23 (1914), 190-210.

Of course, this result does not exclude pointwise convergence. This question was negatively
answered by S. Bernstein in

> S. Bernstein, [Quelques remarques sur l'interpolation](https://eudml.org/doc/158778), Math. Ann. 79
> (1918), 1-12; doi: [10.1007/BF01457173](https://doi.org/10.1007/BF01457173).

For an arbitrary scheme $X=\cup_{n} A_{n}$ of points in $[-1,1]$, there exist a continuous function $f$ and a point $x_{0}$ in $[-1,1]$ such that
$$
 { \limsup _ { n \rightarrow \infty } } \left| L _ { n } \left( f, X , x _ { 0 } \right) \right| = \infty.
$$
The next natural question is the possibility of divergence on a set of positive measure. Such a result was obtained by Marcinkiewicz and Grunwald (independently) for the particular scheme $T$ of Chebyshev nodes :

There exists a function continuous $f$ such that
$$
{ \limsup _ { n \rightarrow \infty } } \left| L _ { n } \left( f, T , x \right) \right| = \infty \quad \text { for all } \quad x \in [ - 1,1 ].
$$
There is an explicit construction of such a function in the book by I.P. Natanson, Constructive Function Theory, Vol. 3, pp. 35-46.

P. Erdos made the conjecture that this negative result holds for an arbitrary scheme $X$ of points in $[-1,1]$. This was proved in

> P. Erdos and P. Vertesi, [On the almost everywhere divergence of
> Lagrange interpolatory polynomials for arbitrary system of nodes](https://users.renyi.hu/~p_erdos/1981-03.pdf), Acta
> Math. Acad. Sci. Hungar. 36 (1980), 71-89 and 38 (1981), 263; doi: [10.1007/BF01897094](https://doi.org/10.1007/BF01897094).

For any scheme of points $X$ in $[-1,1]$, there exists a function $f$ such that
$$
{ \limsup _ { n \rightarrow \infty } } \left| L _ { n } \left( f, T , x \right) \right| = \infty \quad \text {almost everywhere in} \quad x \in [ - 1,1 ],
$$
and the divergence set is of second category. Note that divergence everywhere is not possible just by considering a newtonian scheme i.e. a scheme that repeats points ($A_{n}\subset A_{n+1}$, $n\geq1$, in your notation).

A very nice book about classical interpolation of functions is

> Szabados, J., Vértesi, P., Interpolation of functions. World
> Scientific Publishing Co., Teaneck, NJ, 1990.