I've had a bank account and associated credit card with HSBC in the US for about 5 years now, and I only got an ITIN about 6 months ago. I managed to get the account because they offer a service for Premier customers where you can link accounts in different countries together, and I already had an account in the UK.

However, over the years it's been a massive PITA to maintain since I'm not a resident, and it's getting progressively more difficult - in fact I just moved all my money out of the US to a USD account in a more friendly jurisdiction (New Zealand, where I currently live) because it was getting to the stage where I could see them possibly freezing my account and it being a pain to sort out. I suspect I wouldn't be able to open the account under the same circumstances now.

It's easy to do it through American subsidiaries of Canadian banks like Harris. Not so easy for US-only banks and, as far as I know, impossible for virtual US banks.