I had offered a bounty for the following question that expires in 7 hours:
The results have been very disappointing. No answer was added, no comment, no reaction at all. Which leads me to the conclusion that the limitation of 7 days may not be suitable for a bounty to achieve its goal.
Especially when a question may require some longer thought or when people don't check this site regularly.
I imagine the limitation of 7 days comes from the Stackoverflow.com site, whose activity level and status of professional importance are completely different from Mathoverflow.com. While the 7 days might be appropriate for the former, it may not be so for the latter.
I hope for some consideration towards increasing this limit. With the current 7 days limit (which I regard as a too low for the system to work as intended in some cases), I will not offer any bounty again and will not recommend it to anyone.
UPDATE. When writing my response below to the answer by Joseph Van Name, I have looked at the 6 bounty questions and wrote this answer that the OP found helpful and awarded the bounty, even though my argument was incomplete to solve the problem posed.
Did the system work as intended in this case?
On the surface yes, since it was the bounty through which I have discovered the question.
However, it was a huge coincidence that I had found it, as the bounty was about to expire within a day! Had I looked just 1 day later, I would not have discovered and answered this question, with the bounty of 250 points would have possibly lost. In contrast, chanced were much higher if the bounty stayed for somewhat more reasonable period, especially given the coming holidays.
Now this leads to the question, how many similarly interesting questions are still there unanswered and undiscovered during their short bounty period, precisely because it was so short?
And finally, analysing a bit deeper the effect of the bounty on that question, it is clear that even there it did not quite work as might have been intended. As my answer did not completely solve the problem, it would have been again more beneficial to have the bounty for a longer period, that would attract further contributions to complete the solution.