My husband and I are in the process of divorce. I have a three year old son that is not biologically his. He is threatening to take me to court to get full custody. Is this possible for him to get custody of a child that is not biologically his?

Did he adopt the child?

Sure he can, but he will have to be forced to positively prove paternity via a DNA test if he wants to gain custody.
If you don't fight he has a better chance of winning.

First, ANY question on law needs to include your general locale as laws vary by area.

Second, were you married when the child was born? Is he on the birth certificate?

But yes it is possible, though unlikely.