@Katy *******
The requirements are clearly stated on the Bangkok Bank website. There are several options, one of which is a work permit. One other is a reference letter from an Embassy in Thailand. This actually is a form from Bangkok Bank you take to your Embassy who will notarize the document.
Except for a few branches that deal with foreigners, like the main branch on Silom Rd in Bangkok, most don't have any understanding of the different options and are likely to just say "need work permit". At this point ask them to contact the help desk in Bangkok who will explain to them what is required. Also, you do not need the ATM card with the useless insurance. Just the basic card for 200 or 300 baht. I opened my account on a Visa Excempt Entry.