I've had my PSN account banned back in the PS3 days. There was a thread about it here at the time that received a lot of attention. I got banned because PayPal thought it was suspicious that I bought some PSN funds through the PS Store, and they automatically reversed charges and locked my PayPal account. I wasn't able to log into my PayPal account because one of their methods to prove my identity was broken at the time and I wasn't receiving the text I needed (iirc). So after the charge back happened, PlayStation banned my account. I contacted PlayStation's customer service and sorted it out by paying them back by purchasing a PSN card, or so I thought... They banned me again a week later and refused to give me an explanation when I called. Eventually, word got out from my GAF thread to someone higher up with PlayStation and they sorted it out, unbanned my account, and took steps to prevent that happening to others. During the ban, I was only able to play offline the digital purchases that I had installed on the HDD. I'm not sure if that changed at all since the PS3 days.
If you are buying digital purchases, I highly recommend that you only use PSN/Xbox/eShop store cards and don't use PayPal or credit cards that have the possibility of charge backs. It's also much safer if you somehow got hacked.