When there's a range of delivery dates specified, it often seems designed to show up on the very last day of that range. A significant fraction of the time they haven't even shipped the thing by the first day of the delivery estimate.
There are also an increasing amount of items that say they are Prime but when you try to buy them, it says "it can't be shipped to your address". Sometimes it's understandable, like things that are very heavy or hazardous, like Li-ion batteries. Other times it's completely incomprehensible, like 1/8" pipe fittings. And Amazon customer service can't give any info, they just say "sorry, we're not going to be able to send you that thing."