It is definitely possible, yes. The consultants come from different backgrounds and with different previous experiences.
You would also get through the "usual" case interview process. You could expect, though, to mainly have cases in your sector (bank in your case, if I understood correctly) and to have more difficult cases than the ones you would probably have had, being a newly graduated (e.g., more complicated numbers, more insightful analysis, more broad range of areas where to investigate, ...)
You can also expect to be offered the position of Associate Partner and no higher (I'm talking about McK here), due to the nature of "partnership" that exists in the firm. If you think that this could be the path you like, go for it!