Your question asks, "if everything is possible, is it possible for something to be impossible?" The antecedent of this conditional is "everything is possible" which is equivalent to "nothing is impossible". If the 'thing' that one is quantifying over here is taken to mean any proposition, then one cannot normally say that nothing is impossible, since contradictions are impossible, i.e. in classical logic, contradictions are not true in any possible world.
One could deal with this by restricting one's attention to logically contingent propositions only, that is, those that are neither logical truths nor contradictions. We could then ask, "if all contingent propositions are possible, is it possible for some contigent proposition to be impossible?" Note that this is not the same as asking, "if all contingent propositions are possible, is there some contingent proposition that is impossible?" The answer to the latter is clearly no, but the former is asking only, is it possible that there is an impossible contingent proposition? The answer to that depends on which modal logic one is using. In S5, which includes the axiom ◊P → □◊P this possibility is excluded. In weaker logics such as K and T, this is not excluded, so an impossibility may be possible.
It may help to think of this in terms of relationships between possible worlds, along the lines of Kripke-Hintikka semantics. Suppose that some proposition P is false in the actual world but it is contingent and we are willing to suppose that all contingent propositions are possible, so it is true in some possible world A, but false in some other possible world B, where A and B are both accessible to the actual world. Now we ask, could P be impossible in world B - i.e. is there a configuration of possible worlds under which P is false in all the worlds accessible to B? Under S5 there is not, because in S5 semantics, the relations between possible worlds are reflexive, symmetrical and transitive, so given that A is accessible to the actual world, and the actual world is accessible to B, it follows that A is accessible to B, and hence P is possibly true in B. But in weaker modal logics, transitivity does not hold and one can have iterated possibilities (and necessities) that do not collapse in this way.
So your question effectively reduces to: which is the appropriate modal logic for expressing the idea that anything is possible? Depending on what kind of possibility you have in mind, you may wish to use a weaker logic to allow for iterated possibilities.