I remember calling the time number a lot in the Seventies. Our clocks and watches had to constantly be reset, and the sweet voice of Jane Barbe was the only way to do that. I think the first accurate time piece my family owned was when we got a VCR in the Eighties.
(YourAreaCode) TI6-1212.
As in TIme 6-1212.
Worked in every area code in America.
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I still remember the time number from the 60s (and 70s) was POP-CORN. At the time, I never bothered to actually note the number. 767-2676
Back when they were first putting in digital switches in my area in a small town, I remember picking up the phone to find I was listening to a conversation between two people (no idea how). I kept quiet and listened for a while - pretty mundane. Someone said “do you hear something?” then they carried on until they hung up. Bottom line, I’m sure - when digital switches were first being introduced they had their flakely times.
As to the cost of long distance phone calls - there’s the apocryphal story back when it was bloody expensive, but after direct dial, of the ex breaking in, or had a key, or whatever and while the person was away for the weekend, dialed the “talking clock” in Tokyo or Hong Kong… left the phone off the hook.
What OP is describing is a variation of what was called (as I recall) a tie-line. They were sort of the beginner rung of phone-phreaking back in the 1970s, though I’m not familiar with the exact situation described (connecting by dialing any unassigned number).
Some telcos set up special pairs of numbers in each exchange for use by linemen. I have some pairs, gleaned from someplace, written in my 1970s phone stuff notebook. Seems like they didn’t “supervise,” meaning you weren’t charged long-distance or toll fees to call them. That used to be a big deal, and I remember using one to talk to my best friend (for free, from a motel) while on an out-of-town school trip.
Similarly, there were numbers that you could call and you’d get a busy signal, but you could talk to other callers in between the busy tones. That took the kind of patience only teenagers seeking contact with other nerds could muster, but I remember calling ones where I could make out a dozen or more voices (no long-distance charge, of course, because the line was busy). For some reason, I think those were always in big cities, so they probably were an oddity of crossbar exchanges.
It was TI-4-xxxx in the DC area. Don’t know if TI-6 would have also worked, but allegedly the number was TI-4-2525. Actually after the TI-4 part, any four digits worked. A useful piece of information back in the rotary phone era, because dialing 1111 was noticeably faster than 2525.
And yeah, we needed that number back then. You’d lose power during a thunderstorm, and analog clocks would be stopped at whatever time the power went out, which would be no problem if it came right back on, but often it would be out for a while. And digital clocks, if you had them, would be flashing 12:00, same as now.
Back in the times when you really needed that time recording, you didn’t need to dial an area code; you just dialed the 7-digit local number. Area codes had a 0 or 1 in the middle digit, but local prefixes didn’t, so the switching system ‘knew’ if you were dialing a local number or not.
This worked until people and businesses started needing more lines for fax machines, then cell phones, computers, etc. That created a need for more numbers than would fit in the available area codes, so they had to open up most 3-digit numbers for use as area codes, which meant we had to dial 10 digits for local calls.
In the UK, the “dial back” number was set up for engineers testing new phone installations.
I think any 767-xxxx number would work. I didn’t know about 846-1212 (TIme 6-1212). Maybe there were two different services depending on whether you wanted to hear a male or female voice? 
Was the number of the form 119xx ? When this number rang back, instead of the ordinary ring tone you’d hear ‘beep BEEP beep BEEP beep BEEP’ or ‘BEEP beep beep BEEP beep beep’, etc. Which pattern you got depended on what xx was in the 119xx dialed number. I was told (here at SDMB??) that the facility was used by technicians who came to your house to fix a phone.
Funny: I’m pretty sure that 119xx was the number even though I last dialed it 55 years ago. I can’t remember where I left the book I was reading ten minutes ago.
I can remember that back in the 60s, we used to get what we called ‘crossed lines’ all the time. I would pick up the phone to dial a number and instead of a tone, there would be voices. Sometimes they would sus that someone else was listening, but quite often I was able to eavesdrop. I quickly learned that what people say to each other on the telephone is pretty boring.
Now that you mention it, I remember learning that years later.
For the OP’s situation, almost certainly, it was an accidental “party line” created by the failure of the machinery (and back then, it literally was mechanical equipment) that played the special information tones and the nearly universal message recorded by Jane Barbe.
It eventually “just stopped working” because either someone was bored enough to fix it, or more likely, the central office equipment was being upgraded and the “problem” disappeared once the new exchange was running.
My history with these things was being hired in 1984 or so to make circuit boards for timers that would allow linemen to dial into clean (silent) lines for testing, but disconnect kids hoping to use the test facility as a party line.