Sometimes a person answers a bounty question really quickly and the answer is even accepted.
Due to the following points listed on the set bounties page:
It is your responsibility to award the bounty during the seven-day period or the 24-hour grace period that follows it. You will receive several e-mail and site notifications as the bounty period draws to a close.
You must wait 24 hours before awarding a bounty.
The OP forgets to give the bounty even when he/she had mentioned it. This has happened to a number of people and myself many times.
There are two questions with similar queries:
Feature Request: Send Automatic Email to Users Who Accept Answer Without Awarding a Bounty
Add popup alert box to choose bounty award when accepting the answer on bounty question
My proposal for the same
Once the answer is accepted by the OP, the person who answered the question should get the priviledge to send out a notification to the OP after a day and only one notification per day should be allowed