Thread title says it all.
I remember hearing this phrase ever since my childhood so I’m pretty sure it predates the 80s by quite a bit. My best guess is this is something from the early part of the 1900’s when the middle class started to expand rapidly.
Ability to afford “nice things” + children = no more nice things.
Does anyone know where the phrase came from specifically?
I’ve heard that in OTR in the late 30s. So the phrase defintely does way back
some discussion here THIS IS WHY WE CAN'T HAVE NICE THINGS! - internetmeme meme thisiswhywecanthavenicethings | Ask MetaFilter
Discussion on that site ended in January with no conclusion.
I’ve only heard the phrase in context with “People who break things or can’t be trusted with expensive stuff”.
For example, you might work somewhere that’s been without a microwave for ages, because someone (widely believed to be Bob) left tinfoil on their casserole and short-circuited it.
So, finally, after much pestering to The Powers That Be and maybe some people chipping in a few bucks themselves, a new microwave is purchased so everyone can, once again, enjoy hot meals at work.
Until a few days later, when someone- most likely Bob- leaves the tinfoil on their casserole and short circuits the microwave.
To which you’d explain to someone “This is why we can’t have nice things.”
“Yep. That, or I steal them.” -Cheryl, “Archer”
I will concur that this is the only use I have encountered. We can’t have nice things because of the idiots (someone included in the we) that break/steal/destroy/violate.
I am not sure if this is what the OP was trying to convey. My first reading made me think the OP was saying we can’t have nice things because we can’t afford them.
I think it was my mother. Or, at least, she popularized it. Or, if nothing else, she sure said it a lot.
I agree that it is just a mom thing. My mom said it whenever I broke/ruined anything. Now as an adult, I use it jokingly when somebody else breaks something.
I always thought it started when the kids pulled a black velvet Elvis painting off the wall by accident and the dog peed on it.
I think it was first uttered by the wife of the guy who domesticated the first dog.
I can vouch for its presence in 1940’s New Hampshire among the Irish Catholic Mothers. Well, not personally, but you should see my Dad twitch whenever he hears it. . . 
From Wikipedia:Reference desk/Archives/Language/2010 April 6 - Wikipedia
The line was popularized by Paula Poundstone, who was using it by 1988: “She was one of those angry moms that use to get mad at absolutely everything. I remember the time I knocked a Flintstone glass off the kitchen table, and she said, ‘Dammit, that’s why we can’t have nice things.’” There are anecdotal reports of non-famous people (usually mothers) saying this earlier, so she probably did not originate it. It was further popularized in 1999, when Colin Quinn used it on Saturday Night Live: “American warplanes attacked newly installed Iraqi anti-ship missile launchers along the Persian Gulf. An enraged Saddam Hussein scolded his troops saying, ‘You see, this is why we can’t have nice things!’”
Anecdotal but true: my mother and my aunt used to say this all the time, sometimes for real but usually in jest, because evidently they’re grandmother said it all the time, histrionically, in the 1930s whenever kids would be kids and anything got chipped, broken, torn, or came within 6 degrees of getting so.