If you are going to make up a fake birthday and you think you might have trouble remembering it, picking a nice round number might make it easier.
Eventually, they ran out of valid SSNs for Jan 1st births in some years. (The number is on the form DDMMYY XXXYY, where XXX is assigned sequentially and YY are control digits.)
Hence, for any given date the system can accommodate 1000 people, plenty in a country with some 1000 births a week. Until you start assigning a certain date to people with unknown DOB, that is. They are now assigned a random date.
It's a particular problem in refugee communities, especially where there may be common names. No certificate, often. What happens is that the first time they need to know their birthdate is when they have interaction with a healthcare system, and the doctor (or the admin staff), when told "Oh, some time in 1931, I think", puts "1/1/1931" into the records.
All it takes is two of "Samuel Goldstein, born 1/1/1931" in the same suburb and you've got a serious risk of misidentification when one of them has a heart attack and turns up in an ambulance. Misidentification of patients might be relatively uncommon, but the danger when it happens is severe.