That is no longer true. It no longer requires a wise child to know its father—a well equipped lab will do. Paternity testing is the stealth biotech, a technology that, unlike more newsworthy competitors such as human cloning, is now well established, reliable, and in common use.
So far, the most notable consequence has been to sharpen the three-way conflict between women with babies, men who don't want to support them, and welfare departments that want someone other than them to pay the bill. In the old days, the mother and the welfare department could convincingly argue that the husband—more recently the lover—was the father, and so owed duties of paternal support. They are now in the uncomfortable position of trying to claim that a husband who is provably not the father of his wife's child—who is therefore most naturally described as the victim of his wife's marital fraud—is still obliged to provide child support, and similarly in cases where the relationship is less formal than marriage.
The longer run implications are more interesting. From a technical standpoint, it is now possible to combine any mating pattern from strict monogamy to complete promiscuity with assured paternity. How many of those options actually go into common use will depend, among other things, on how much of our sexual behavior is hardwired and at what level.
If, for example, male sexual jealousy is itself hardwired by evolution—as a mechanism to make sure that men don't waste their scarce resources supporting other men's children—nothing much can be expected to change. Men will still have a strong preference for sleeping with, and having children by, women who are their exclusive mates, and the likely result is something close to conventional monogamy. If, on the other hand, evolution has simultaneously provided men with a desire for assured paternity and a taste for promiscuity—both of which make sense from an evolutionary point of view—we may end up with a form of group marriage, or some less structured alternative, becoming common.
To some extent this has already happened, driven by a slightly older technology—reliable contraception. We already have a society where a level of female pre-marital sexual activity that would have been considered scandalous in most past societies is widely accepted and widely practiced—because it only rarely leads to unwanted children. The implication of the newer technology of paternity testing is that a similar pattern is becoming possible for reproductive as well as non-reproductive sex.
Readers interested in what might—or might not—be the cutting edge of such developments may find the alt.polyamory web site of interest.