^ nope, one part of the Captain's test, is to show the testors your bankai. so that means all captain that entered through this way must have bankai.
Another way to become a Captain is by recommendation, so I'm not sure if that methods still requires you to have bankai. Therefore, assuming most Captains entered through the first method, they all must have Bankai. The only known Captain who doesn't have Bankai is Zaraki (that we know of) because he became a Captain by defeating the previous Captain in one-on-one combat. Since all you gotta do is win, one does not need Bankai if one wishes to become a Captain through this method.
The only non-Captains with Bankai are Renji, Ikkaku and Ichigo. Renji, isn't a Captain cause either a) he doesn't meet the OTHER requirements of becoming a Captain [other that have Bankai], b) there isn't enough time right now for him to take the exam, get the paperwork done, go through the ceremony of appointing Captain, and getting him a new Haori [Amagi doesn't count cause he is a filler character, and fillers are full of bull], or both [probably]. Ikkaku doesn't want to be Captain, so he keeps his Bankai a secret so that no-one will pressure him to take the test, and Ichigo is just a Substitute Shinigami.
Although there is something I noticed I suppose is worth mentioning: Maki Maki exclaimed that there were FOUR Captain-level Shinigami around when Tousen, Komamura and their VCs turned up to face the 11th Div. If Maki wasn't mistaken and the translation was correct, that would mean Hisagi and Iba have Bankai.
EDIT:Only realised how long it is after I posted.. gonna edit to make easier to read.