I ordered something from Thinkgeek once. It got from New Jersey to Montreal in two days, and it took them a week to actually get it to me. They'd no-show on days customer service told me they'd deliver, then they'd deliver on days customer service told me they wouldn't be showing up. That was the day I learned the value of shipping things to work.
I had a replacement MacBook power adapter shipped out. They left a notice once, and then customer service told me that there'd been three failed attempts and no further attempt would be made. When I asked where the package was, the agent didn't seem to know, but guessed that it was at the depot. When I got to the depot I gave my delivery slip to the guy behind the counter, who came back and said he couldn't find the package. I told him to go look again and this time he actually found it.
I ordered a kit from 23andMe. DHL tried to deliver it once and then returned it to sender. I got them to send me another one (at a $25 fee), which DHL never delivered. I tried a third one, and this one was "at the depot". I went there and waited an hour in a small cement room with nothing but a chair in it while they looked. Eventually they came back and said they'd given up.
Every single encounter I've had with DHL has shown they're completely incapable of performing the core aspect of their business: getting a thing from one person to another. Worse still, they don't really seem to care or even understand that a screw-up is happening.
Various articles in the media state that he does read them, and typically forwards to the relevant executive in charge.
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.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.