As Francesco mentioned, some of these timelines differ a little by geography. At BCG in the US, you are a consultant 24 to 30 months, and PL 18 to 24 months if memory serves - so in 4.5 years you can become a Principal. Hard to be elected Partner before 8 or typically 9 years though.

Also, re. the number of spots available... I think it is easier to get a position created if you are lucky to be in a fast-growing office. As they say, a rising tide lifts all boats -> faster growing companies & offices will give you more opportunities.

Different companies (and offices) also have different degrees of "meritocracy" - i.e. how quickly it's possible to get promoted. While I was at the London office of Bain & Company they consciously went from people occasionally being allowed to get promoted one period early (each period is 3 months) to a fully meritocratic system where people were promoted as soon as they were able to demonstrate they were ready.

It's worthwhile bearing in mind when you consider which companies you are applying to, but bear in mind you'll be joining a group where every individual thinks they probably have a chance at being better than average.