Hmm.. somehow some lines have been crossed somewhere...
I'm an American and worked as a financial advisor in Munich for 3 years.
Ketchup is completely right in that you need to give your SS number and fill in a W-4 to allow the bank or fund company to report your earnings to the IRS. This applies to all "allowed" investments (keep reading...).
The actual regulation I believe DDBug's bank is talking about, however, is that banks and investment companies are not allowed to sell American citizens or foreigners who are residents in the USA any investments that are not
approved for sale in the USA. Example: as an American you can very well buy shares of the Templeton Growth Fund, Inc. in Germany, however, you cannot buy shares of the very similar Templeton Growth Fund (EUR). Because, you guessed it, the sister fund in Euros is not approved for sale in the USA.
I suppose the words "can" and "cannot" here are also misleading: the point is that the banks or investment companies are not allowed to sell them to you. If you have, say, an online depot... well, who's to say what you can or can't buy...

And as long as you report your interest/capital gain earnings in your yearly tax filing, I would think there shouldn't be any problems.
This regulation went into effect about 2 years ago. It holds for funds; buying individual stocks is not a problem, as well as individual bonds and other interest-bearing investments.
If you have an investment account already, try asking your bank for an online option so you can trade whenever you like; get your advice elsewhere! If you don't already have an investment account, you can try to open one either online or at a bank and hope that that particular institution isn't very nitpicky... Or open an online investment account in the States and see the thread about cheap money transfers...
Good luck!