@Todd **********
Good point. You can try but may not be possible. I suggest Krungsri Bank (yellow) might be the best to try. However, you will be applying for your Non-O in Thailand so as soon as you get that you would be on a long-term visa then and should be able to meet any Thai bank's requirements. You can apply for your in-country Non-O when you have 30 days to go on your current permission to stay. If granted it would be granted immediately and you would be able to take that and open an account at a Thai bank. A caveat that it's not always easy to open an account in Thailand for a foreigner, especially an American (due to US government policies imposed on foreign banks). But with persistence, it can be done. Pick large branches that you have reason to believe may have existing foreign customers (in tourist areas, for example). I recommend Krungsri and Bangkok Bank.