I concur with Seriati. You look at the large multinational banks, and they're easily going to have the resources to add the software and process to be compliant, and they are likely to serve rich Americans with lucrative accounts. I have no trouble believing that the French equivalents to local credit unions are choosing to avoid the whole mess.
The BBC article doesn't give many details, which is frustrating. They state that "banks would rather refuse" and other vague statements - they don't even name the bank that "Jane" was denied by. Fabien wasn't actually refused, he just faced a paperwork nightmare, which is also not great.
All that said, one might easily be led to conclude this is widespread, otherwise why was renouncing citizenship the last resort, as described by Jane?
This article talks some about this action. They cite ABN AMRO as closing portfolio investment accounts help by US citizens. AMRO is an almost exclusively Dutch bank, and its an investment house - not a basic checking account.
Interestingly, the Patriot Act is cited as a reason why Americans overseas can't just open a US-based account in many cases.
There is a fairly elegant proposal being made to fix this, which is a same-country exception. So, if you can prove to the bank in Brazil that you reside in Brazil, the bank would be exempt from reporting. I'm sure there are ways to game that system, but on the face of it - it works. People hiding money in the Caymans don't want to live there.