I think Cannie could. Or it would work with Cotherine. But imo you don't get
Connie out of
Catherine unless your accent makes the "a" & "o" sound alike (
Isabel /
Isobel).
In fact it could work in
France, where the nn rules are very permissive. I know a
Lea (leh-a, "eh" like "men") nn
Lili, and though my first name is
Claire some friends call me Cici (si-si).