There certainly could be merchants who were very wealthy in the Middle Ages, but the conditions for the Medici's success were made easier by the political system in place in Florence at the time.

I actually wrote about the Medici on a number of occasions:

This answer explains how the Medici were able to pull the strings behind the government of Florence, while this other answer explains the government of Florence more in-depth.

I also participated in this discussion examining the Medici's rise, and this other discussione examining the transition from the Republic of Florence to the Grand Duchy of Tuscany.