You can report a problem with your order and get a free month of Prime when they miss their dates (up to 12 per Prime renewal). It's somewhere in the shipping guarantee. If you paid for any accelerated shipping they'll refund those fees.

I used to be able to nearly double my Prime length this way. The shippers in my area have gotten much better though recently and I haven't had a late package in almost a year.

Edit: Hm, they may have quietly discontinued it recently. It used to be listed on this page:

  https://www.amazon.com/gp/help/customer/display.html/?nodeId=201117450

And I can find articles up to December 16 stating it still works. Doesn't hurt to try though, they're usually pretty accommodating.

The guarantee means that if the package doesn't arrive after that point it's their responsibility to assume it's not going to arrive and make alternate arrangements.

In other words, if they "guarantee" tomorrow at 5 PM and it doesn't show then it's late, whereas if they "estimate" tomorrow at 5 PM and it doesn't show then it's whatever, call back tomorrow.