I think there's a little distinction to be made here between interacting with a C-level manager and advising him or her.
In terms of interaction, it is generally up to partners / directors to talk and present to people at that level. If you just graduated from Business School you should expect little to none direct commication with CEOs, CFOs, COOs, etc. There are obviously exceptions, the most common that comes to my mind is when a client is relatively small and C-level managers end up interacting with consultants on a daily basis.
That being said, when working on a project you WILL be providing advices to a company and they will go all the way up to the senior management team. Even though you probably won't be the person talking at meetings and steering committees, the slides presented will be the result of your job and you might need to comment specific details of the analysis or of the results. In my opinion, this can be very satisying and ends up being a very valuable "training" for when you will progress in your career.