I thought I had discovered a random bug in Femap, but I was mistaken...
Femap 11.3 introduced the very useful feature that Mesh Points on a surface would be automatically split onto their appropropriate split surfaces if the original surface was split using the Geometry Editing in the Meshing Toolbox.
This new feature was working superbly until we discovered that it was randomly not working.
Anyway, the functionality works perfectly when the original surface is meshed. It does not work when the original surface is not meshed. So, if you allocate Mesh Points to an unmeshed surface (eg. unmeshed to help visibility for spot welds, rivets etc.), and the surface needs to be split, then first put a mesh on the surface so the Mesh Points split properly - and delete the surface mesh later if useful.