Edit for clarity: Particularly Western/ American society in the 19th-20th centuries. When establishing paternity would have legal ramifications, but no way to conclusively prove anything.
Hi OP. As this question stands, it is so non-specific that it's effectively asking about the entirety of human history, worldwide, which means that it falls afoul of this sub's example-seeking rule.
Can you narrow it down by specifying a legal jurisdiction / region / country and general time period? Eg 19th-20th C USA, or other. That information will greatly assist anyone considering answering here. Thanks