Student or Learner
I've heard some teachers say she's terrible with names but some say terrible at...
5jj is right, but you can't generalise that beyond names.
I'm terrible at chess/dancing... (Not 'with')
I'm terrible with women/animals... (Not 'at')
So, it's more like
at + things/activities you're bad at, say I'm terrible at Maths/sports....
with + things/activities is not so direct, say, I'm terrible with names = I'm terrible at memorizing/recalling the names of my students.
Is that right?