There are a few ways.
1) (Both Corporate/Personal) : Open a bank account locally with a bank that also has branches in the USA - then use their international banking center to facilitate the process of necessary documents at your local bank (I.e. HSBC, Citibank offer this service) - http://www.citibank.com/ipb-global/homepage/newsite/content/... , http://www.expat.hsbc.com/1/2/
2) Just personal: Open an "High Net Worth" account. Also can be done from anywhere in the world, but you must usually keep at least $30K USD in the account liquid in order not to be charged hefty fees. Citibank offers these accounts as well (see link above)
I am in the process of trying this and it has become much less simple. Certainly for HSBC a remote account opening is considered "next to impossible" - it requires above branch level sponsorship from the bank which in practise means turnover 1m+. I am told US anti terror laws have made a big impact on opening non resident accounts in past few months
I am hoping I can workaround the problems but my bank managers expectations are low - I may be visitin the states sooner than planned.
What trouble are you having with HSBC? I've just spent about an hour on the phone with them to open an account in Australia and all I'm required to do now is go to a Local Branch near me with my ID / documents - the account will be ready in a week, two weeks max.
2) Just personal: Open an "High Net Worth" account. Also can be done from anywhere in the world, but you must usually keep at least $30K USD in the account liquid in order not to be charged hefty fees. Citibank offers these accounts as well (see link above)