Disney owns Winnie the Pooh characters.

The entertainment giant built a multi-billion dollar franchise around the characters created by A. A. Milne, and the legal question is if they owned it. Looking at the fact pattern, the author, A. A. Milne, assigned the rights to the Schlessinger family who then assigned to Disney. So Disney would own it, and the courts have upheld its ownership rights.

Did the Schlessingers ever have a case?

It would depend if they got the rights to assign copyright or not from the original copyright holder. I would assumed they have an "all rights" agreement, which would mean yes. If the rights are separately listed and it doesn't include the right to assign, then no.