Particular incidents that didn't seem believable (spoilers)
In the episode that Bradley Whitford wrote, the one where Josh leaves the administration to become campaign manager for Santos, the administration is having difficulty in Congress because of the opposition of a particular senator. After several attempts, the president himself meets with the senator. We see and hear them arguing, but nothing is resolved. Then at the end we're just told that the measure passed Congress. How did the president manage to convince him? We're never told. Very dissatisfying.
In the episode following Toby's admission that he is the leak, CJ refuses to talk to him, but instead immediately calls in the lawyers. That's not very believable. CJ is not a lawyer and would not have known that she shouldn't do that. In fact considering their long working relationship it would have been more believable if she had made some kind of commentary on it.