I'm new here, but I have had several experiences with this. Back in 1997, a doctor in NY City had his lawyers threaten a Major Hotel (where I worked) with a lawsuit if they did not correct there caller ID. It turns out, AT&T was sending the number out on all calls going out on a plain old T1 route, no Dchannels, with one digit off in the exchange portion of the number. People from all over the world with caller ID or Call Return thought they were calling back to the main number of the hotel. The Hotel is 2200 rooms and all calls outside of NYC went this route. I confirmed the incorrect number was coming only from the AT&T trunks.
AT&T apologized and corrected it immediately. I don't remember if the doctors name appeared to the callers.
I'm a field tech now and a recent problem came up with the names that appeared on the caller ID. An attorney's office has every partner, 39 of them, listed individually in the White Pages by name, under the main number of the firm. They were fighting with AT&T over an attorney's name that appeared on the caller ID and not the name of the firm when somebody called a clien in Long Island. AT&T passed the buck to verizon, as janaya stated, the local carrier is in control of the number also controls the listing. Verizon said everything is fine on their side. If I called my dad on his verizon telephone line, the law firm name appeared correctly in capital letters, no problem. I called my wife, in the same village, on a number that is ported to Cablevision OptimumVoice, and an attorney's name appears. This was the major complaint I was working on. What a coincidence, the person who reported the problem also has Optimum Voice. I have a second line at home, same exchange, but ported to Verizon Voicewing service. I called my wife on the Voicewing line and a different attorney's name appeared, this time in upper and lower case text! The customer tells me they had a couple of instances with this name too!
I didn't get to follow up on the resolution, but I gave the customer situations the could be recreated to prove out with Verizon.
Before I arrived on this trouble the customer had both AT&T and Verizon override their outbound CLID on their circuits with the one and only main number they wanted everyone to see. That was done and I was able to prove it with the dchannel messages printing on all trunks. The main number was delivered no matter what the switch sent out, and no matter what carrier, the listed name depended on what LEC received the call on the terminating end.
The only company that handled the firm name and attorney names is Verizon so it fell back on them.
I hope this clarifies who is able to do what for you, with or without ISDN/PRI.