Speaking from experience, yes (this was 9 years ago though and may have changed due to stricter laws).
I lived in London and was moving to NYC and wanted to have a functioning bank account upon arrival. I banked in London with HSBC and asked them if they could set me up. They connected me with the right people in the US and after many forms I had a fully functional US bank account with a foreign address and without having a social security number - and I was (am) just your average person. You will most likely not be able to get a credit card through them because of lack of credit history (unless you are ridiculously rich or go for a secured credit card), but a debit card should be possible.*
My advice is to talk to your local bank and see if they can help you, although it will help if they operate in the US.
*I have heard from various expats that American Express may be willing to issue cards in the states based on their existing relation with clients in other countries, but I digress. If you have an Amex in Switzerland or Sweden I would recommend talking to them.