The popular version says you choose any two of the three letters. That reads as if there were three equal options: CP, AP and CA. There are not.
A system spread over a network does not get to decide whether partitions happen. Cables, switches and cloud networks fail. And to the other nodes, a node that has stopped answering looks the same as a broken network. So P is not a feature you pick; it is a fact you live with. The real choice is between C and A, and you make it only during a partition. A so-called CA system is one that assumes the network never fails, which in practice means a single machine.
Brewer said this himself in 2012, in an article titled "CAP Twelve Years Later": "The '2 of 3' formulation was always misleading because it tended to oversimplify the tensions among properties." He also made the other point: partitions are rare, and "there is little reason to forfeit C or A when the system is not partitioned." A well-built system gives you both on a normal day.
There is one more trap. The choice is not one switch for a whole database. Many systems let you choose per request. In the etcd test below, the same node refused one kind of read and answered another, a second later.