The SSDI is based on a self-reporting mechanism. If you acquired the birth certificate of someone who had never acquired a social security number or whose death had not been reported, it shouldn't be in the SSDI.
Thus, while it might be hard to verify whether or not someone is in the index, there are many people who are not in said index. Consider: why does someone's death get reported to Social Security? So someone else can claims the death benefits. There wouldn't be much incentive to do that for a child.
That being said, the older you are, the harder it would be to explain why you never had an SSN, making this much harder to pull off today.
Update: That being said, it appears that the US switched to a national data collection system in the 1990s. It's the NVSS (National Vital Statistics System, http://www.cdc.gov/nchs/births.htm) which will track the births and deaths for you.
Thus, while it might be hard to verify whether or not someone is in the index, there are many people who are not in said index. Consider: why does someone's death get reported to Social Security? So someone else can claims the death benefits. There wouldn't be much incentive to do that for a child.
That being said, the older you are, the harder it would be to explain why you never had an SSN, making this much harder to pull off today.
Update: That being said, it appears that the US switched to a national data collection system in the 1990s. It's the NVSS (National Vital Statistics System, http://www.cdc.gov/nchs/births.htm) which will track the births and deaths for you.