It is not a verified fact, but it is believed that the 24 hour timer on a tile is triggered upon the ''attempt'', and '''not''' when/if you actually obtain an item from it. Therefore if you find nothing, you are better off trying another tile instead of repeatedly trying on the same one.
It is also been commented on that tiles that are not foraged or botanized for long periods of time (a week) have higher chance of getting something than tiles that are checked 24 to 36 hours apart.
And don't forget that you can both forage and [[Botanizing|botanize]] on the same tile, these actions are on seperate timers.