I got a bank account w/o a visa (on my passport waiver while waiting for my visa). You need to go to a branch that does a lot of business with foreigners. So look for branches in tourist and expat hotspots and go during the week when they're not that busy.
Branches who do not have many foreigners walking in are simply not interested in dealing with the hassle that comes with foreigners.
Are you looking for something to solely to satisfy Thai immigration or looking for something to both Thai immigration and cover you in case of ill health?
There are much cheaper option for the former. (Off hand my policy is < $400 per year. ) They're not terribly good for the latter but healthcare cost in Thailand is super cheap compared to the US anyway.