SIPC is a corporation - a legal entity separate from its owners. In the case of SIPC, it is funded through the fees paid by its members. All the US brokers are required to be members and to contribute to SIPC funds.
Can it go bankrupt? Of course. Any legal entity can go bankrupt. A person can go bankrupt. A country can go bankrupt. And so can anything in between.
However, looking at the history of things, there are certain assumptions that can be made. These are mere guesses, as there's no law about any of these things (to the best of my knowledge), but seeing how things were - we can try and guess that they will also be like this in the future. I would guess, that in case of a problem for the SIPC to meet its obligation, any of the following would happen (or combinations):
Too big to fail - large insurance companies had been bailed out before by the governments since it was considered that their failure would be more destructive to the economy than the bailout. AIG as an example in the US. SIPC is in essence is an insurance company. So is Lloyd's of London.
Breach of trust of the individual investors that can lead to a significant market crash. That's what happened in the US to Fannie Mae and Freddie Mac. They're now "officially" backed by the US government. If SIPC is incapable of meeting its obligation, I would definitely expect the US government to step in, even though there's no such obligation.
Raising funds through charging other members. If the actuary calculations were incorrect, the insurance companies adjust them and raise premiums. That is what should happen in this case as well. While may not necessarily solve a cashflow issue, in the long term it will allow SIPC to balance, so that bridge loans (from the US government/Feds/public bonds) could be used in between.
Not meeting obligations, i.e.: bankruptcy. That is an option, and insurance companies have gone bankrupt before. Not unheard of, but from the past experience - again, I'd expect the US government to step in.
In general, I don't see any significant difference between SIPC in the US and a "generic" insurance coverage elsewhere. Except that in the US SIPC is mandatory, well regulated, and the coverage is uniform across brokerages, which is a benefit to the consumer.