No really true and I have confirmed this through multiple contacts with Amazon when they failed to ship a few thousand dollars in Hurricane supplies. From the Prime page
“If the item you’re ordering is out of stock or unavailable to ship immediately, the shipping method time starts when the item ships. For example, it will take two business days after an item ships to reach you with Two-Day Shipping.”
The Amazon manager agreed that if the item is in stock that means they will deliver within two days. Every time they have failed to do this they extended my Prime membership 1 month with a simple phone call.