If you open a brokerage account nowadays, almost all will ask for a W8-BEN form certifying you are not a US person. For US persons (citizens, greencard holders, or substantial presence in the US), you would fill out a W-8 form.
There are three possible outcomes (ranked from most to least likely):
1. The bank tells you to get lost
2. The bank allows you to open your account, but bans you from almost all mutual funds and any access to US markets.
3. The bank allows you to open your account and access US markets, but bans you from most non-US mutual funds.
[Update: In Spain, there would be possibility #4, the bank manager has no fucking clue and lets you do whatever you want.]
The reason for this is simple: in order to comply with the US FACTA rules, the easiest thing for a bank to do is just certify they have no US persons as customers.
The reason the banks ban you from non-US mutual funds is that the mutual funds have the same problem: they need to certify that none of their investors are US citizens if they want access to the US market. Also, they can't offer their product to any US clients without potentially running into problems with the SEC, which they would also prefer to avoid.
Basically the US doesn't want US citizens owning investments in the US via an a foreign brokerage without knowing that these are US citizens. This is due to the fact that the US wants to be a great destination for foreign capital (without taxing interest or capital gains), but wants to make sure the US citizens don't try to circumvent this by investing in the US via a foreign company or bank.