It depends on several factors:
- Role (e.g. it's much easier to pass the interview applying from a top MBA)
- Company (McKinsey PST may require a lot of prep)
- Prior background
- Time you can dedicate per day (if you are working full-time it may take much longer)
Purely case prep requires a couple of months with dedicated 2-3 hours per day.
Also, it's often more about quality rather than quantity:
- Don't count the cases you've done yourself
- Don't count the cases you've done from the casebooks
- Count the cases you've done with experienced candidates (50+ cases solved / passed the first round) who can give you a real case and a good feedback.
In that case, you need 40-50 cases depending on the role, office, etc (e.g. MBAs in top schools need just a bit of practice compared to regular candidates. Both because of their knowledge and because the on-campus hiring is not that tough)
If you take a good coach, you can make it in 20-35 cases. The coach will give you the right knowledge. Partners will help you integrate the skill. At the end of the day, it's just a skill that can be trained.
Why do you even need partners and coaches? When you are stressed during the case interview and out of energy having the 3rd case in a row, 90% of your questions and reactions should be fully automatic. Otherwise, it's extremely hard to solve the case, trying in find out what you've memorized from v.Cheng book.