Saying x out of a possible y, where y is a number that normally doesn't take an article, is quite common. It is idiomatic and correct. It is not informal or slang. It happens with any adjective, not just with possible.
It is correct to say one out of four, but not * one out of possible four. If I were to hazard a guess, I'd say the reason is probably that possible is an adjective, which can normally only be used to modify a noun (a possible cause), while four is not a noun but a cardinal number, which modifies a noun itself (four camels, one woman) and is not preceded by an article. But some cardinal numbers are also nouns, like a hundred, a thousand, etc. That's why it is quite unremarkable that we say a possible hundred. When the need arises to turn a lower number into a noun, because we want to modify it with an adjective, we use the article just as we do with a hundred, to form a possible four. In other cases, there is no need to nominalize ("nounify") because an article would be redundant, as in * I see a one horse.
I believe one level out of four is short for one level out of four levels; adding possible would give one level out of four possible levels. Shortening that would give ?one out of four possible, but that is somehow not preferred with adjectives (it is common with (fossilized) participles: one out of four remaining). For some reason we prefer turning four into a noun and putting the adjective before it, calling up the article a as above.
Something similar is probably behind we came back in twos: when the word two is used to mean a set of two, i.e. a noun, we may want to pluralize it. That is normally not possible with two, since it isn't a noun; but we can turn it into a noun and add the s in certain circumstances.
There are also personal pronouns, which function much the same as nouns, expect that they normally cannot take articles, adjectives, or cardinal numbers: * Two they left my house is impossible. But sometimes we want to specify how many things or persons are referred to by a pronoun. Because a personal pronoun can be the object of the preposition of, as in part of him, several of us, that is how we do it: two of us left, after the two of you. For that we turn the number two into a noun. As such it is modified by a prepositional phrase (of us, of you) and can take the definite article the.