They were so stupid as to not even be able to get the months of the year right:
September (Septi-) = 7, not the 9th month!
October (Octo-) = 8, not the 10th month!
November (Novem- = 9, not the 11th month!
December (Decem-, Dec-) = 10, no the 12th month
But one small example of the preponderance of the evidence to the contrary.
That’s because the new year didn’t begin in January back then, but in early spring. There have been various systems, and the calendar has been adjusted several times. Our current system, the Gregorian calendar, came into use in 1582.