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.
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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.
Good luck!
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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.
money.stackexchange.com/users/830/Korneel Bouman
There are several major US banks including Bank of America, citi and Banco Popular that will open an account for people without a SSN.
Most will require an in branch visit to open the account .
As some one else mentioned American Express will open accounts in other countries based on an existing relationship or at least they used to.
money.stackexchange.com/users/1873/stoj
Can you get a bank account in the US?
Yes.
Can you get a bank account with a specific bank in the US?
That depends on the bank.
There's more paperwork involved, and not all banks offer the service.
I would guess that the big retail banks (US Bank, Bank of America, JP Morgan / Chase, et al) are most likely to, but they are also the ones with the worst reputations for customer service.
money.stackexchange.com/users/1739/Benjamin Chambers
See this website.
In my opinion you should physically exist there to open your account.
The bank needs to fulfill all requirements such as checking your identity, taking your signatures for future transactions etc.
However, there might be some exceptions as Banking industry works pretty much on personal relations and money power.
Also check these links: http://answers.google.com/answers/threadview?id=722141 http://askville.amazon.com/open-bank-account-abroad/AnswerViewer.do?requestId=7004217 and http://www.talkgold.com/forum/r18761-.html
money.stackexchange.com/users/None/
In addition to the other answers, Harris Bank (now owned by BMO) allows Canadians living in Canada to open accounts, perhaps they consider other countries as well.
They have excellent customer service.
money.stackexchange.com/users/1375/Matthew Read
I had to open a bank account in the US without having the right paperwork initially (SSN really).
All the bank asked me to do was fill in a W8 form in lieu (instead) of the social security number.
money.stackexchange.com/users/210/Peter K.
Note that there are two very different issues here.
Opening a bank account is one thing, opening a credit card account is quite another--you're effectively beyond the reach of any enforcement action, why would any bank lend you money?
money.stackexchange.com/users/1884/Loren Pechtel
I'm Swedish, but living in Switzerland at the moment. Can I open a US bank account? Either when in the US on vacation or through the internet/phone?  There are lots of situations when I find myself in need of an american credit card or just a bank account. When ordering things online (most retailers refuse to accept my orders if I try to pay with my Swedish or Swiss credit card, even if I ship it to an US address... ) etc etc.
Can I open a US bank account even if I don't live in the country?
money.stackexchange.com
<united-states><credit-card><bank-account><non-resident><europe>
money.stackexchange.com/questions/5028
money.stackexchange.com/users/2213/Erik Tjernlund
It is possible to open a US account without an SSN. If your local bank operates in the US as well, the might be able to help you. It is generally agreed, no by no means certain, that you should be able to open a bank account in the USA without being there in person. However, the exact answer will depend on the bank in question.
It is generally agreed that you can open a bank account in the USA without being in the country. Whether or not you are able to open an account will largely depend on the bank. There is doubt as to whether or not you can open an account without being physically in the USA.