Your maximum hand size is checked only during the cleanup step of your turn. At any other time, you may have any number of cards in hand.

If multiple effects modify your hand size, apply them in timestamp order. For example, if you put Spellbook (an artifact that says you have no maximum hand size) onto the battlefield and then put Doctor Octopus, Master Planner onto the battlefield, your maximum hand size will be eight. However, if those permanents entered in the opposite order, you would have no maximum hand size.

The number of cards you ultimately draw may be affected by replacement effects. For example, if you have six cards in hand as Doctor Octopus's last ability resolves, and you also control Thought Reflection (an enchantment that says "If you would draw a card, draw two cards instead"), you'll end up drawing four cards.