According to the staff report in http://www.straightdope.com/mailbag/mphonedial.html, AT&T was the only phone company in the 1960s. That’s funny - I could have sworn that mine was supplied by the GPO (General Post Office (of Britain)).
There’s a couple of hundred countries outside of America y’know.
The phrase in question is “AT&T (the only phone company at the time)”. However, in the United States in the 1960s there were numerous small independent telephone companies, most commonly in rural areas. So this would have been more accurately written as “AT&T (the dominant U.S. phone company at the time”.
I did a quick google search on “independent telephone companies”, and according to the site below, “there are more than 1,300 independent telephone companies operating in the United States, companies that never were part of the former Bell System.”