"Most Winningest" Regular Season

There will be a total of $\binom{10}{2}=45$ regular season games. If we want the most teams to win seven games, we can start with the floor when $45$ is divided by $7$ . This is $6$ . However, if $6$ teams win $7$ games each, that leaves $3$ wins among $4$ teams. With these $4$ teams alone, there must be $\binom{4}{2}=6$ wins, so this is not possible. Now, we can try $5$ teams. This leaves $10$ more wins among $5$ teams. As $\binom{5}{2}=10$ , this works. Thus, we can have $\boxed{5}$ teams that can make the playoffs.