No, you do not have to go to the top 20 universities to be an investment banker. I have heard people from other universities get in as well. For example regarding London universities, investment banks I think have extensively recruited from London Southbank University and London Metropolitan University. This shows that someone not in a top 20 uni has equal chances. Also, it does not depend so much on the uni, but on the overall profile, which should include valuable work experience at top financial institutions, extracurricular activities etc.