@stephybabes92 I think the issue is that the organizations are school-sponsored organizations that are related to the distinguished degrees each university offers. They're not like clubs, which are Simsverse-wide, they're tied to one particular university.
Since your Sim is studying Physics at Britechester and not at Foxbury, where there's a distinguished Physics program, she can't join the Robotics or Brainiacs Organizations because those are only for Foxbury students and she's a Britechester student. Ditto for her housemate studying drama at Foxbury; Britechester offers organizations related to their distinguished degrees so the Art Society is for Britechester students.
This is one of the reasons distinguished degrees are better. The larger reason is that distinguished degrees offer a bigger career boost.
I'm assuming your Sims didn't get into the distinguished degree programs. In short, it's kind of a penalty for not getting into a distinguished program rather than something you're doing wrong. If you had an A in high school and have some related skills, you can study Physics at the University that specializes in Science and has organizations related to science, and if you don't get in you can still get a degree but it's not as big of a boost and doesn't come with those organizations. (And ditto for Drama and the University that specializes in the Arts.)
@stephybabes92 , I agree with @xochiquetzl_xkvn about the distinguished degrees. I think that is why your Sim can't join the robotics club because Britechester doesn't offer a distinguished physics degree, only a regular one. 😉
