I have read the too big to fail Wikipedia and as far as I understand, a company is too big to fail (and thus will receive government bail-outs) if it's failure has a significant impact on an important industry or the economy as a whole. Is my understanding correct? For example, a company such as Snapchat would not be bailed out because while it is worth a lot of money, its failure does not hurt the United State's infrastructure or the economy as much as GM's failure.
In economics Too Big To Fail (TBTF) can have slightly different meaning depending on what research you are looking at but generally speaking literature seems to agree that what matters is how interconnected or systemically important firm is (e.g. see Bernanke 2010; Zhou, 2009).
Systemically important firms are such that other firms critically depend on them. In fact in recent literature you will often see people arguing that TBTF is a misnomer and better terms would be “too complex to fail” or “too interconnected to fail” (see Kaufman, 2014), because really TBTF has much less to do with actual size than interconnectedness of a firm.
This is the reason why you usually hear the term TBTF being applied to banks or financial institutions. Financial, institutions are an economic equivalent of infrastructure as they help to allocate capital to its most efficient uses and many other firms depend on them (see Mishkin & Eakins Financial Markets and Institutions).
Consequently, whether General Motors or Snapchat are TBTF depends on how interconnected are with the rest of an economy. If there would be significantly many 'influencers' and 'content creators' that they would become significant part of an economy snapchat might be TBTF. This being said although I was not able to find specific papers on examining how interconnected GM or Snapchat are I am going on a limb to say that probably neither of them are TBTF. As argued by Strahan (2013) TBTF in most cases really applies mainly to financial sector due to reasons already mention above.
In fact Strahan specifically argues in that paper that non-financial firms, using General Motors as an example of non-financial firm that would not be TBTF, can only be TBTF in rare cases. Although the paper makes just analytical argument for this and draws only on general literature without actually examining networks of General Motors specifically (importantly note just because firm is bailed out that does not mean it was TBTF - every company has incentive to argue its TBTF to get bailout and policymakers often cannot examine networks in real time so they have to make an educated guesses about whether firm is actually TBTF or not - if a non-financial firm claims to be TBTF its good to be skeptical given literature but I would not dismiss it out of hand either).
Human beings rely, like the body does, on how resources are allocated within a nation. We can divide resources into essential and non-essential. When one company monopolises an essential resource then we say it is too big to fail. Because were it to fail, then that essential resource is no longer allocated to the disruption of the nation's life.
Of course this is a situation that is ripe for abuse since, generally in liberal economics, it's the competition with rivals that brings corruption down to manageable levels. When that particular 'check and balance' no longer works then there is a possibility for abuse and in neo-liberal economics, where profit is all, we must assume they will pursue that abuse.
This is the reason for anti-monopolistic laws such as the anti-trust laws in the US; or the Glass-Steagall Act that cordons off merchant banking from retail banking. With such laws, no company or corporation should be too big to fail. In fact, the anti-trust suit against Bell Telecom broke up the company into 'baby bells' which allowed one or two to fail without there being systemic carnage.
That is the theory - and for the practise: unfortunately, and in many ways foreseeable, during the neoliberal era such rules and regulations were gutted, because profit-seeking was all, with the financial carnage and human cost we saw in 2008. A depression that bore comparison only with the Great Depression of 1929.