It occurred to me that I had hit the limit on the number of Bluetooth connections on my phone and that this was four.
I tested this by using the Nordic app nRF Connect. It detected all six buttons simultaneously, showing the advertising interval and showing that they were all bonded. From this I conclude that my phone can support at least six Bluetooth connections.
I then activated three iBeacons and these also were detected at the same time, bringing the number of simultaneous connections to nine.