My bank has always made it very clear that online bill payments take 2 days to show up. A tiny amount of forethought takes away all the issues with that

Plus my bank has a same day payment feature, but it costs extra ($11 dollars maybe, I've never used it). When I paid by mail, I always had to pay at least a week before it was due to make sure that it wouldn't be late, and often I paid as soon as I got the bill if I had the money.
I was late with a payment once, because I had gone out of town without considering what would be due, etc. They got it one day late. It was a Macy's credit card bill, so I paid in person as soon as I could. I called customer service immediately, from my car as soon as I made the payment. The representative looked at my record, saw that I had never paid late before, and waived all the penalties.