It depended on when and where you are talking about. Slaves had different rights in different colonies/states over the period 1609-1865.
By the early 1800s, many of the slave states had made it illegal for slaves to own any property without the permission of the slave owner, as exemplified by Section 56 of the 1833 Alabama Slave Code. You'll also notice that in Section 30, the code prohibits freed slaves from living in the state. Once you were freed, you had thirty days to leave Alabama on your way to a free state.
The laws were similar in many of the slave states, forcing free black people to leave the state anywhere from 30 days to one year after being granted their freedom. Many of the slave states went so far as to prohibit ownership of any property of any kind by any slave, even with the slave owner's permission, effectively making slave ownership impossible by a free black person.
The majority of slave ownership by black slave owners in the United States, it would seem, came from South Carolina and Louisiana, who made it a bit easier to live there as a free black person, Louisiana particularly. 3000 black slave owners lived in New Orleans. The vast majority of those "slaves" were the owners' kin. Louisiana slave law had come down from French law, so slaves had guaranteed rights they didn't have in other states. They had the right to marry, and splitting up married couples or children from their parents was prohibited. This and other factors led to more situations where a free black person could "buy" a relative into freedom in Louisiana than elsewhere.
However, outside of New Orleans and Charleston, the number of black slave owners are pretty miniscule, and of the ones there are, they tended to have more than a financial interest in their slave(s), i.e. they tended to be family members or friends, just like in New Orleans.
For example, in 1860 in South Carolina, outside of Charleston, there were 69 black slaveowners in the entire state. That same year, in the entire state of North Carolina, there were only eight. (Source.)
So, yes, it was possible, it did happen, but it was very rare. And as rare as it was, quite a number bought their "slaves" for non-financial reasons, but rather to free spouses, children, or other relations.