The maximum amount of stable slots any one character can have is sixteen:
Five slots for having 240.0+ taming skills, nine bonus slots for the three taming skills at levels 100.0, 110.0, and 120.0 plus 2 bonus slots for being
SA upgraded.
Stable slots are based on current skill level
including jewelry. A character with 99.0 skill in one of the three skills can get an additional bonus slot by wearing a piece of +1.0 skill jewelry in that skill. When the jewelery is removed, the creature will remain in the stable.
You can check your number of filled/total stabling slots by the command "stablecount".
Use of the stable is a key factor in "
bonding" a pet to an owner.