Go ahead and overwhelm yourself with all the information we have.
You do love that shift key, don't you - but is it hiding the fact you are unsure about capitalisation in English??
If you are willing to waive your rights to banking secrecy under Switzerland's agreement with the US tax authorities, there are several banks that will open an account for you with as little as 15K CHF. As long as the funds are legitimate (and you don't mind a bit of double taxation) you should have no problem.
You will, however, need to come to Switzerland and be prepared to prove your identity. A US passport will suffice, but you will also be asked your SSN for tax reporting to the IRS. Unlike in the US you do not need to bring "proof" of your SSN, such as the SSA card or a W-2 form, but you should know that the bank will file a W-9 to verify it with the IRS.
(Aside: under the QI program the swiss banks get an EIN for purposes of filing a W-9. This despite the fact that they are not an "employer", which kind of gets to the meat of the moral issue regarding double taxation of earned income. But don't get me started.)
Once the bank has your SSN you will of course be subject to other scrutiny such as Dun & Bradstreet, Equifax, credit reporting agencies, etc. Because of the nature of the KYC regs I can't be too specific as to what we use (I work for a swiss bank), but rest assured we have ways of knowing who you are (e.g., if your uncle is a foreign dictator or you are suspected of serious crimes). Oftentimes Google is all we need to disqualify a customer or at least seek senior management approval.
If you have a substantial amount to invest (i.e., more than you would feel comfortable putting into a suitcase), you should bring statements from whatever firm holds these funds.
If you do not wish to keep your accounts secret from the IRS, you should forget about it -- swiss banking secrecy is not what it used to be, although there are still legal ways to remain somewhat private (such as opening a foreign trust rather than holding the account in your name).
If you can get a residence permit (even a simple L permit for 90 days) and an apartment (NOT a hotel) you can open your account online through swisspost postfinance. The documents needed to finalise the account will be waiting in your mailbox when you arrive, and you don't even need to go to the gemeinde before going to swisspost (they're even open on saturdays).
Sorry for such a long post, but there is a lot of misinformation out there (and a lot of scam artists IMO). One of these days I'm going to write a guide and put it on wikipedia.
I have yet to verify this with my banker friends, so take this with a grain of salt. Depending on what rate they withhold it may be an acceptable tradeoff should you wish to remain private. I should find out later today.
Gawd there are some dodgy comments here.
The limit to open a bank account in Switzerland for a non-resident who wishes to protect his capital and earnings from the unwelcome attention of the NAZI facist IRS is set by the bank.
Therefore, whilst UBS quotes 250,000 CHF minimum and Credit Suisse 200,000 CHF there are other banks with other limits. In addition, the limits are not set in stone, for example, if you are 25, turn up with 50K and say you want to open an account, and there will be another 50K in six months time. UBS / CS will take it from you. Because in 2/3 years you will be at their minimum and accelerating way past it.
As you are a “Us” citizen, you should only consider a bank with zero presence in the US or US administered locations or even under US sphere of influence (e.g. Cayman). USA PATRIOT calls for certain Ferderal activities that you are hearing about with UBS now and UBS will buckle like a 16 year old after two bottles of wine. So you don’t want a bank with business or income derived from any other country, and particularly your home country.
If I was a wealthy American with money to spare, I would go to Singapore and open a bank account there, not only is it cheaper, the earnings are higher and the secrecy in Singapore will exist long after Switzerlands has been eroded.
As to the second payment, it is either to the SBA or the EBK and I believe it is a trivial amount (i.e., 1%).
I want to correct the earlier comment above about finding a bank with no US presence -- this is not enough. You must ensure that they have not signed a QI agreement with the US, and AFAIK all members of the SBA and LBA have signed this. I know for a fact that LGT asks all US clients to complete a W-9, regardless of whether they live in the US, CH, or wherever, and regardless of whether the account is opened in ZH or in Vaduz.
The problem with Singapore is that there is a govt. regulation that you must have $2M SGD to invest (approx. $1.6M USD). Failing that you can go to one of the more dodgy tax havens such as Panama.
One thing you might try is to open a "salary account" with one of the cantonal banks, because I have heard that some of the smaller banks do not require a residence permit and they will certainly not ask you for a W-9 as the purpose of the account is not for investment. I suppose one could open an account for 5-10K CHF, and there's nothing stopping you from making further deposits except that you'll only be getting 1% interest!
Now for the soapbox -- the $1.6M USD requirement for Singapore means that upper middle class Americans are unfortunately priced out of the market for private banking. Swiss secrecy is dead for Americans, and don't let anyone tell you different. It's a shame, because in my opinion undeclared assets and banking secrecy are the only ways to keep world governments honest. As we say in the US, "the higher taxes are, the fewer people pay them". If voting at the ballot box doesn't work, you can always vote with your feet (move abroad) or vote with your dollars (secrete money abroad to get out of the clutches of the IRS).
As US taxes are scheduled to rise quickly in 2010 (and beyond that who knows given the impending collapse of Social Security), it's no surprise that the US gov't is getting tough with the Swiss despite the presence of a Republican administration. If you think it's bad now, wait until Obama is in office!