Consider a couple special cases of multiple Bernoulli trials: one where $p$ remains fixed, and one where $np$ remains fixed as $n \to \infty$ ($p$ represents the weighted probability of the trial, and $n$ represents the number of trials).
We have a few theorems that help us convert this setup into some well-known distributions. In the former case, we have the de Moivre-Laplace Theorem which leads to a normal distribution; in the latter case, we have the Poisson Limit Theorem which leads to a Poisson distribution.
Consider the first scenario, since your question appears to want to talk about approximating continuous distributions with Bernoulli trials (if you want to talk about discrete distributions, the discussion is not much different). In such a case, the de Moivre-Laplace Theorem gives us a fairly easily understood way to obtain a specific continuous distribution from discrete trials. But this is pretty far from approximating every distribution in such a manner.
Norbert Wiener used the Cameron-Martin Theorem to show that we can approximate arbitrary distributions with a normal distribution under the condition that the target distribution has finite second moment (equivalently, that it is mean-squared convergent). Later work extended this to other well-known distributions: uniform, beta, gamma, exponential, etc. Likewise, discrete distributions can be approximated in a similar way.
So, in a sense, we can definitely approximate -- or derive in a limiting sense -- mean-squared convergent distributions from Bernoulli trials, given some conditions on $p$.
I am not sure what happens when we remove this restriction. I would imagine it would be difficult, if not impossible, to approximate some distributions that do not have finite second moment (e.g. the Cauchy distribution) in this way.