Let's go through a thought exercise.
Let's say there are 6 American conglomerate banks remaining after this financial crisis: Citigroup, Bank of America, JP Morgan Chase, Morgan Stanley, Goldman Sachs and Wells Fargo. There are more big, important banks, but for simplicity and name recognition, let's stick with these.
In a world with no restriction on too-big-to-fail:
Let's say Citigroup, because it's dumb, takes on a whole bunch of risks that all go kaput at the same time as the result of a systematic shock. It fails. The FDIC, funded by the US taxpayer, needs to bail out all of Citigroup's depositors. Citigroup's failure casts doubt upon its ability to pay down its derivatives and loan obligations. Holders of these obligations - the other 5 banks - need to write them down, sending them into a liquidity crisis as they struggle to raise more capital to fill the holes in their books. If they can't raise the capital, they fail, sending their own derivatives and loan obligations into question, and further burdening the survivors. Eventually, everyone dies.
In a world where too-big-to-fail is regulated:
Each of these 6 banks is split into 2 smaller banks. Any one of the new banks is not big enough to take down the financial system; none of them are "too big to fail".
However, everyone still has loans outstanding, and generally there's quite a bit of overlap as to the groups they're lending to. If the loans of the original big banks are split among the small companies, then systematic shock happens, and then instead of Citigroup failing, Citigroup A and Citigroup B both fail (or Citigroup A and Bank of America B, or whatever). You're in the same situation, then, because the same number of loans failed, they were just split among two banks who could only handle half the size of failure as the original big bank - the other 10 banks scramble to find liquidity and the process happens.
In other words, if banks need a 2:1 ratio of capital to loans, it doesn't matter if there's one bank with 40 in loans, 20 in capital who loses 6 in capital and needs to find 6 vs two banks with 20 in loans, 10 in capital who each lose 3 in capital and need to find 3 each. Systematically, small banks and big banks pose the same collective risk.
There's an argument that opening the prospect of an individual bank failing will prevent stupid risks to begin with because each individual bank can't be sure of a bailout. This has two problems. Firstly, in the above scenario, every small bank except the very first one or two to fail needs a bailout, also. So you don't really help the system that much.
Secondly, it ignores the fact that firms are not living, breathing entities; they're run by humans who get fired. Maybe everyone knew the government would bail out the banks and let them live. Most of the chief executives, however, have left in disgrace (the exceptions being the ones who did a good job - Blankfein, Dimon, etc). Say what you like about contractual compensation commitments (and ignoring the fact that the government has been abrogating them) - all of them would rather be doing their job than not doing it. Does it soften the blow and make risk more palatable? Probably, but it's not as if they thought it'd be consequence-free. Certainly there's something to be said for toughening the Boards of Directors at public firms to make them act with some teeth towards CEOs - both strategy and compensationwise. Done by a privately elected board of directors, this can actually work in the shareholder's interest. Regulation isn't going to solve that very well, though - it's too "one size fits all".
An alternate "split up" method is to separate the different financial activities that happen within a bank - for example, commercial banks and investment banks and insurers couldn't be under the same roof. Splitting commercial banks and investment banks would exchange frequency of bailouts for severity. You'll still have "too-big-to-fail" institutions within each sphere. What that ends up doing is creating more frequent crises that are less severe. If the investment banking industry has a crisis, then the investment banks need to be bailed out. If they're linked to commercial banks, however, the commercial bank may provide the diversification necessary to make regular equity capital or debt capital work, without a bailout. Thus, large and diversified banks fail less often. Of course, as we've seen, when they do fail, they fail spectacularly because there are so many moving parts that fail all together. Would more frequent but less severe crises be better? Possibly, because they'd derail the economy less often. However, given that the US underinvests so badly, sustained periods of good times actually aren't a bad thing if they prompt actual investment, whereas more frequent crises may undermine investment further. Without actual research, this isn't an easy question to answer.
What's the real solution to "too big to fail"? Create an orderly system of quickly dismantling failed banks so you don't send banks that haven't failed scrambling for liquidity. That'll help, but not solve the problem, by lessening the level of required writedown to the amount by which the failed bank fell short instead of some prospect of more.
A better idea would be to stop politicizing loans; if Fannie and Freddie hadn't started buying up mortgage assets at the order of Congress, the bubble would never have been as big in the first place. The same goes for small-business loans.
However, ultimately, while good practice (by government and individual firms, as well as consumers) can lessen the risk, you can't legislate it away. It's fundamental to fractional reserve banking, and unless you wanna take that away (and send us back to the 1840's, stunting American growth forevermore), it will always be a risk. We're better off creating a method for dealing with it next time than trying to eliminate it altogether.