Over in the US, they're in the second week of a government shutdown - an event so regular in their dysfunctional, ramshackle "democracy" that it has a name and a list on Wikipedia. And in NZ, curious people are asking the obvious question: can it happen here?
The good news is that no, it can't. The simple reason is that under New Zealand's Westminster system, "supply" - funding the government - is what constitutionally determines who is in power. So if a government is unable to do that, we don't have a shutdown, we have a coalition renegotiation or an election instead.
The more complicated reasons have to do with budget processes and incentives on government. The US has these shutdowns because they dick around with the budget process, engage in brinkmanship, and engage in other childish bullshit. One of the reasons they do it is because they have weak party discipline, and consequent weak accountability to the public. Another is that the US's strict separation of powers means legislators have no investment in the success of executive government. A shutdown isn't their problem, but a problem they make for someone else. Here, the executive is drawn from the legislature, so the governing parties (in theory a majority of the House) are invested in success. The ability of the government to continue to perform its basic functions is very much their problem, to the extent that threatening to interrupt it is like holding a gun to their own head. And they can't just refuse to do it or run out the clock like they do in the US: McGee's Parliamentary Practice in New Zealand (practically a quasi-constitutional text) says that "The Government cannot avoid asking the House for supply, for to do so would be to abdicate its responsibility as a Government." While I don't think it has ever been tested, if they piss about, fail to do their jobs and pass the usual bills for interim funding and allow current budgetary approval to expire, then its treated as a loss of confidence and an election or coalition renegotiation automatically follows.
Which brings us to the ultimate safeguard: us. US politicians play these games because the US electorate tolerates it. While it has never been tested in practice, I suspect the tolerance of the New Zealand electorate for such bullshit is very, very low, to the extent that any government which threatened such an interruption (let alone presided over one) would be voted out on their arse. US politicians appear to face no such threats. And you can see the result of that in the media every January.