I discovered today while reconciling our bank statement that I accidentally paid the Visa bill twice last month. It was a pretty hefty bill, actually, because there were two sets of plane tickets on it and miscellaneous other stuff.
But (and this is the happy part) nothing bad happened. We didn't bounce any checks or anything. I've confirmed that the credit card company agrees that we paid twice, so we currently have a credit on the account and won't need to pay anything this month. Yes, we've missed out on a trivial amount of interest this way, but it's clear that it's not worth the time to sort this out.
I am terribly glad that our finances have gotten to this level of comfort.