The bullets on a player's character card indicate how many cards the player can hold in his hand at the end of his turn. This is the player's hand size limit.
A player can have more cards than he has bullets, but he must discard any excess cards at the end of his turn.
If a player loses a life point out of turn, he does not discard any excess cards until it is the end of his turn. If the player uses a card to draw or steal more cards, he does not discard excess cards until the end of his turn.