Yes, there are a number of them. As everywhere else, it's not perfect. Depending where you come from and what you're used to, they will vary. I will have to tell you my own:
The main downside coming from a European is the isolation. You don't realise really how FAR AWAY from everything you are. The closest country is a 4 hour flight (Australia, and this is if you fly to the East Coast... flying to Perth takes 7 hours), and apart from the Pacific Islands there no other place to go without embarking on an 8 hour flight minimum, and that's to South East Asia. You simply cannot pop out to visit family and friends. Any trip to Europe/America is at most a yearly thing that costs a huge amount of money and leaves you seriously jet-lagged both ways. And you will waste at least 2 whole days of your holiday just with flights. Living in New Zealand is not for the family-oriented who need to be close to loved ones, unless you bring them all with you.
Another dowbnside for me is the lack of social activities after hours. In the country I come from, the streets are buzzing after 5 pm, all shops, cafes and restaurants are open, ther are people in the street until past midwnight every day. in New Zealand, everything closes after 4. Even in bigger towns, most cafes close at 3 or 4 pm, shops close at 4 pm, and there's barely anyone in the streets after that.After 20 years I still miss the liveliness of European cities in the evenings.
Another downside for me is the inequality. New Zealand is a colonised country, and the indigenous population (Maori), as all other colonised populations in the world, suffer from displacement and lower socieconomic conditions which leads to deprivation, poor health outcomes and higher crime in that demographic. In general, crime in New Zeeland is very low and the country is very safe. However there is an underground New Zealand where domestic violence, drugs, child abuse, incarceration and multigenerational deprivation are rife. Not only Maori families, but disproportionately in maori families. Together with that comes a low-level (I mean low compared to other countries where I've lived, included my own birth country) racism by the white (pakeha) population which is umcomfortable to see, as many New Zealanders fail to have the insight to consider the privilege the colour of the skin and background brings. In my previous career I lived and worked with provoleged people around me, so I didn't experience this in first person. As a midwife giving care to all kinds and shapes of women and families, it becomes very obvious. I went through a period of my life when I felt very ashamed that I had not been aware of this.
Something that for me is not a big deal, but for many it may be, is how geologically active New Zealand as a country is. It is called "the shaky isles" for a reason. New Zealand is laying onto a series of deep faults. I went through the Christchruch earthquakes, which killed 185 people in only one city, but I have also felt earthquakes strong enough to wake me up everywhere I have lived, and they are expecting a massive one in the next 30-50 years. There are also volcanic eruptions (yes, a number of them while I have been living here), tsunami alerts regularly, floods, a cyclone here and there (mostly in the North island those ones) and every now and then someone's house/garden
/shed in Rotorua gets swallowed by boiling sulphuric mud. It's something accepted by Kiwis and whoever wants to live here: you need to live with it, because this is not going to change any time soon.
Apart from those, I can't really say there are many more downsides for me. New Zealand is one of the least corrupt countries in the world, so that was a breathe of fresh air for me. Most people are incredibly honest, you can travel around as a solo woman traveller feeling safe, I leave my mobile phone on the table outside when I go to order another coffee in a cafe, and the only time I forgot my bag with my wallet and everything in a cafe, it was returned to me with everything in it a day later. The feeling of safety is one of the main reasons I feel at home here. There is nothing like the feeling of mostly trusting your fellow humans.