At McKinsey in North America it works like this
You must join the firm as a BA (no other starting role as far as I am aware allows this) and then stay in the firm for 2 years. Additionally, you must maintain a certain rating during your entire time at the firm, this could include being promoted to Associate or EM early but is not necessary to avail of the sponsorship. The firm then pays for your MBA or any other two year graduate program! (there are 2 different financial methods to do this depending on the tax implications for you).
If you do want to take advantage of this, you have to commit to coming back to the firm for 2 years post your MBA in order to consider the debt paid off.
This differs significantly in other regions, and also McKinsey no longer considers an MBA a requirment to move up in the firm. All BAs that are performing well automatically get promoted to Assocate after a certian amount of time
Hope this helps,