Hey Huw :)

To cure your confusion: No, its not that my drill press doesnt have enough power to drill that hole. Its about the flutes of the drill and the chips which you have to get out of the hole while drilling. With increasing cutting depth the chips get stuck and so you have to get the drill bit out of the hole to clear it. And the possibility of the bit getting broken inside the hole increases with cutting depth.
Of course if the diameter of the drill bit is big, you can (with enough power) drill a deep hole without breaking the bit.

Now my confusion :D : To cut an M16 i have a regular single thread cutter bit and a wrench. Im not the strongest guy, but i often cut M16 by hand and the required momentum is moderate, i would say. So either your cutter is dull, or something else is wrong with it

Best regards

Maik