You can never rely on information on certificates to be completely true, there is always a chance of being led up blind alley.
A lot of people never had their full birth certificate, there was not the call to produce it as there is today. For example, my father only had a shortened version which simply gave name, date and place of birth. A full certificate was an unnecessary expense for many people. My father travelled to Canada and back in 1910 without the need for a passport.
Even my late mother in law who born illegitimately in 1920 only had a shortened version of her birth certificate, that was all she needed to obtain a passport to visit us in Malta in 1963. I once asked her what her mother's name was, she said she had no idea as she had never seen her full certificate , it was only about 30 years ago when I obtained her full certificate that she learned who her birth mother was.
On her marriage certificate to my husband's father the space for her Father's information is blank. When she married her second husband she gave her foster Father's name rather than admit she was illegitimate. So anyone researching her would find it impossible to find a birth in that name.
No one would have imagined that well over a hundred years later family historians would be unearthing family scandals.