I can find no reference to the CABE named before the 16th Century in England, Scotland or Ireland. However, there are many recorded in Europe, particularly France, mostly contained in the Aquitaine area (pronounced CABAY) which m aked me think that they could have been Huguenots.
In England, the earliest recorded CABE is a Henry CABE born in Suffolk in 1559 which would coincide with the Huguenot refugees arrival. The early CABEs appear to be grouped mainly in London, Suffolk and Devon, all points of entry for protestant refugees from continental Europe. Many also went to North America. My own line appear to originate from the Devon area.
One train of thought is that they were from the McCabe sects in Scotland and Ireland, at some time dropping the prefix. However, the lieklihood of them all changin is remote as most would have been too proud of their heritage to do that.