Realisitically, this has not been the solution implmented (in the EU & US, at least). In the EU, it is even more crucial as the "solutions" to this problem are applied to state finances as well as financial institutions.
In terms of policies, there are two competing approaches: (1) Reduce the size of "too-big-to-fail" institutions. (2) Regulate them more heavily (or some other strategy) so that they will not fail. In the EU, this is being applied to states, not just financial institutions. Rules that (supposedly) reduce catastrophic risk.
Almost all seripous policy proposals are in the no. 2 category. Tighten regulation, reduce the risk of failure. Tighter regulation lends to stronger incumbents and larger average company size so by doing 2, you are probably doing the opposite of 1.
As I said, I don't know what Bernie's proposal is or how mature it is as a policy (as opposed to a politician statement). It would be notable if a left wing politician propsed loosening bank regulations, though definitely not impossible or unreasonable.
That seems like the most reasonable response. And yet, since the great recession, our policy has been "make 'too big to fail' even bigger".
The problem is that the banks have become too powerful for anyone to challenge. A Teddy Roosevelt type of political leader can't exist today.
Standard Oil was as powerful as the US Government in that era, as were the railroads. JP Morgan was far more powerful than the US Treasury. Cornelius Vanderbilt - pre Roosevelt - all by himself had greater financial capability at his peak than the US Government at the time.
The difference, is back then there was wide-spread and growing fear of the combinations and would-be monopolies. Today, Americans are relatively unconcerned by Microsoft (desktop), Google (search, android), Walmart & Amazon (retail), Intel (microprocessors), Facebook (social), Cisco, Boeing, etc.
Seems like it's fitting here, does it not? Certain banks being "too big to fail" is an idea passed among persons within our culture.
That should be the priority.