Ben and Mabel (Mable) cardgame? What's the "real" name?
This sounds like a form of Contract Rummy.

There are dozens of variations of Contract Rummy which have different rules of how many and what size of sets and sequences are required in each successive round. In my brief overview of different variations on pagat.com, I didn't find an exact match to the clues you've given, I suggest you have a look there and try different contracts!