I have two. I'm an Executive Assistant, so relationship and compatibility are very high on my agenda.
A very well known financial services company that the agency repeatedly told me I was lucky to get a foot in with. I would be working for two people who were based in the US and had assistants there but also each spent a week a month in the UK.I wasn't sure anyway, but had two really great interviews with various people and thought I could make it work and find a niche for myself. Came down to the actual person I would work with and he cancelled twice. Then they just offered me the job, without meeting him at all. I turned it down as I need to work with someone invested in me and the working relationship, he said he trusted the judgement of those who had met me. The agency could not believe it. I just said he clearly didn't value the working relationship and that I could not decide to work so closely with someone who I hadn't met.
Second, I actually accepted the job, but then left after 4 weeks - not proud, but I quietly packed up one Friday, emailed them to say it was not working and I had a personal issue and was abroad. The agency really pressured me to accept.
Interview 1: with HR who spoke almost in a whisper, I am very gregarious and loud and assertive. Told the agency no, they convinced me to second.
Interview 2: with the director I would work with. He was pleasant, but I am the sort of person who needs a click, or something about them - charisma, arrogance, eccentricity, but he was just bland. I said no, they again convinced me to third.
Interview 3: call with CEO in another country and his EA. Again went ok, but just a bit dull. Said no.. you get the picture.
interview 4: lunch with remaining members of the team (small team so prudent to see fit). The most painful lunch ever where I just went round the table asking people how long they had been there, what their weekend plans were.
Said no, got offered and agency pushed hard for me to accept.
Got there and it was the dullest place ever, so quiet that I would leave the room to make a phone call. The Director had said he wanted someone who just did things and didnt feel the need to check everything with him, then he questioned every single decision I made, from restaurants, to times of cars to the pens I bought for the office.
Moral of the story - trust your gut and instinct.