Say if I have a MineBot and also wanted *insert pet here*.
Is it possible to own 2 of them?
Or maybe if one is asleep, could I still use another till that one falls asleep and then switch back and forth? Of course they'll be woken up when needed.
The only real way to have 5 without expanding the pet inventory would be to have the minebot as one, since the minebot can be stored in the bank or fashion shop inventory. You wouldn't want to try 5 otherwise because you wouldn't be able to unequip it to extend its life with an herb, much less to level or enchant it.
Since you can expand the pet inventory to 80 slots, and the minebot can be stored in the bank/fashion shop, 81 is the theoretical max. But we don't have 81 pets yet. ^_^
BTW, if you have all 4 pet slots full you can't dart, and you can't open anniversary boxes or fortune cookies. You have to have at least one slot free in every inventory type to use those.