Can I go to a Mechanical Engineering graduate program with a BS in Neuroscience?

The OP mentioned in other threads being at UCR, specifically in the humanities and social studies division.

UCR’s humanities and social studies division has the highly unusual policy that students cannot switch out to a different division after the first year: https://chassstudentaffairs.ucr.edu/transition_advising/index.html . So even if a second year student is fully eligible to change to a natural science or engineering major by those divisions’ requirements, s/he cannot change out of the humanities and social studies division. I.e. the OP’s issue may not be that his/her new major or division is “full”, but that his/her current division will not let him/her leave.

If you are thinking that “this makes no sense”, then you are correct.