I have a BSc and MSc in Mechanical Engineering and after 1 year working in the field, I got fed up and switched to McKinsey: one of the best decisions in my life.
I was too much fact-based and rational in my cover letters and CVs. But that's more for technical positions, in the consultancy world you have to "sell the dream". It took me 5 months of case studies (to be fair, I studied only on weekends because I was working) to switch to that kind of mentality. Your CV will say which your achievements are, but the cover letter has to shout your motivations, the perception has to be that MMB (or the consultancy firm you are applying for) is the job of your life.
Drop me a message if you need a piece of advice, I'll be happy to help!