Can we say anything about the distribution of the sum of not iid random variables?
There are many versions of Central Limit Theorem; iid versions tend to be taught because they're relatively easy to prove (e.g. if the MGF exists, that leads to a reasonably simple demonstration of the CLT). Because there are many CLTs, we should be careful of saying "the CLT" unless the one we mean is clear from context.
There are versions of the central limit theorem for cases where the variables are not iid and specifically there are versions where you have independence but the distributions differ. Basically, individual variances can't be too large relative to the rest (in hand-wavy terms, individual variances have to be vanishing fractions of the total, the formal condition depends on which CLT you're looking at).
For example, see the Lyapunov CLT
There are also cases of the CLT for situations where dependence exists (see later in the same article for some examples).