Names in English typically have three components. A given name, middle name, and surname. Within any given surname, assuming people choose only to pick from the top 500 for the given names and middle names, that is 500 Γ 500 combinations.
However, there is in fact more variety in given names. The top 1,000 given names account for 72% of children according to the US Social Security Administration.
So there are actually at least 1 million viable combinations. Certain enough that except for all but the most common names ("John William Smith"), a person's trinomial name is almost already going to be unique.