Yes, I had the phone in my hand... but I found the problem! The guy at DTAC in Tesco Lotus said "Oh, you have turned off the Internet!" Not his exact words, but that's what he meant.
So he clicked various things and the "4G" icon at the top of the screen started to show activity.
I got him to add some money to my account anyway, while I was there, so I haven't actually tried to "Refill" again, but I'm fairly sure it will work now.
If it doesn't, I'll post back here!