What I'm thinking of is a statement like this:
A member of x institution is bad, therefore x institution is bad.
A member of x institution supports y, therefore x institution supports y.
This seems to be similar to 'tu quoque' fallacy, where an argument is discredited because the one arguing it does not follow the argument's conclusion them-self. However, I'm not sure whether it can be assumed that there is an argument being discredited, nor that the fact of the institution's existence implies an argument that the institution is good.
This also reminds me of the genetic fallacy, but it seems like it is the opposite. As far as I can tell, the genetic fallacy says "we can't trust Tim because he is from Venus," whereas this says "we can't trust Venusians because Tim is one and we can't trust him."
I'd like to be certain. Is this a fallacy? If so, which one?