I'm in agreement that "possible" is binary. Things are not more or less possible than others. "Better possible" is outright strange, though I can see how someone might erroneously infer it from "best possible", i.e. "the best possible solution to the banking crisis." But in that case, we parse the sentence as "best (possible solution)", not "(best possible) solution", if you know what I mean. "Best" isn't modifying "possible".
If the probability of something is being compared, it's more or less likely, or more or less probable. You only use "more"/"less" to modify a yes-no, true-false property if your name is George Orwell and you want to show us how all animals are equal, but some are more equal than others.