Hi Nanu,
It was often the case of infants who were poorly at birth with a possibility of not surviving, that they were quickly and privately baptised, if they survived there was a procedure where they were "received into the congregation". I'm not quite sure what form the receiving into the congregation took, probably something along the lines of a full baptism in front of the congregation. I did see a post somewhere of a suggestion that perhaps wealthy people may have used private baptisms, how true that is I have no idea.
Regards