If A and B are independent events then the probability of the two events is P(A)*P(B). So P(A+B)=P(A)*P(B) and even if both probabilities are 1 (certainty) their product can never exceed 1, so P(A+B) can't exceed P(A) .
If B depends on A we know that probability must lie between 0 and 1, so if B depends on A, even if P(B)=P(A)=1 (certainty) their combined probability cannot exceed 1, because there is nothing beyond certainty.
Example: if two cards with identical backs, one white and the other black, are placed face down, P(A) is the probability of first picking a white card=1/2. P(B) has one card left to pick only, so P(B) is either 0 or 1 and their combined product P(A+B) is between 0 and P(A).
[The situation is different for P(A or B), because this results in P(A)+P(B)...]