No cap.
However, depending on the route you're playing there's certain characters you can't recruit.
You cannot recruit:
On Golden Deer: Dimitri, Hubert, Dedue, Edelgard.
On Black Eagles (Crimson Flower): Hilda, Dimitri, Dedue, Claude.
On Black Eagles (Silver Snow): Dimitri, Dedue, Claude. (You are able to recruit Hilda but only have the moon before timeskip to be able to.)
On Blue Lions: Edelgard, Dimitri, Dedue, Hubert.