To answer the original question in this thread, I have an old copy (1926) of the Book of Common Prayer and under 'The Ministration of Publick Baptism of Infants' it says
'When there are Children to be baptised, the Parents shall give knowledge thereof over night, or in the morning before the beginning of Morning Prayer, to the Curate. And then the Godfathers and Godmothers, and the people with the Children, must be ready at the Font, either immediately after the last Lesson at Morning Prayer, or else immediately after the last Lesson at Evening Prayer, as the Curate by his discretion shall appoint. And the priest coming to the Font, (which is then to be filled with pure Water,) and standing there, shall say
Hath this Child been already baptised, or no?'
I think we can judge from the language that this must have been custom for some time. Also, you will find more than one family baptising infants on the same day especially if it's a feast day or the saint's day for the saint the church is dedicated to.