Here is what one sees after unraveling the usual “Leray‐localization’’ (or purity) argument: one reduces the finiteness of
H^2_et(X, ℤ/n)
to the finiteness of the zero–th cohomology of a certain sheaf of “wild–vanishing–cycles’’ supported on the fibres of X over the primes dividing n. Concretely, if you set j∶U=X\bigcup_{p|n}X_p and Z=∪_{p|n}X_p, then from the localization triangle

 j_! j^*(ℤ/n) → ℤ/n → i_* i^*(ℤ/n) →[+1]  

one gets an exact piece

H^0(X, R^1i^! ℤ/n) → H^2(X, ℤ/n) → H^2(U, ℤ/n)\,.  

Now U is a scheme over Spec ℤ[1/n], so n–torsion is invertible there and Milne’s (or Deligne’s) finiteness theorem tells you H^2(U, ℤ/n) is finite. Thus one is forced to understand

H^0\bigl(X,\;R^1i^! ℤ/n\bigr)\,.  

But a point y in Z lies on a fibre X_p with residue characteristic p∣n, and one checks by purity that the stalk

(R^1i^! ℤ/n)_y  ≃  H^1\bigl(\Spec(O_{X,y}^{sh})\setminus\{y\},\;ℤ/n\bigr)  

is exactly the group of continuous homomorphisms
π_1(\text{fraction field of }O_{X,y}^{sh}) → ℤ/n,
i.e. it is Hom(K^×/(K^×)^n, ℤ/n) where K is that local field. But as soon as p∣n the group
K^×/(K^×)^n
is {\em infinite} (because the residue field of O_{X,y}^{sh} is infinite, the 1+℘–part gives an infinite pro–p–quotient), and hence each stalk of R^1i^! ℤ/n is infinite. There is no reason a priori for the global sections of such a non–constructible, infinite–rank sheaf to be finite.

In fact nothing in the literature gives you a uniform bound on the “wild’’ part of R^1i^! ℤ/n, so one cannot conclude that H^0(R^1i^! ℤ/n) is finite. Equivalently, one expects that in general

H^2_et(X, ℤ/n)

can be {\em infinite} once n is not invertible on X and dim X≥2. All of the standard finiteness theorems (SGA 4½, Milne’s Arithmetic Duality, Deligne’s constructibility of Rf_! for ℓ≠p, etc.) require the torsion to be prime to the residue characteristics.

— In particular one does recover finiteness of H^2_et(X, ℤ/n) in all the “ℓ–adic’’ cases (i.e. when n is invertible on X) and of course when dim X=1. But for p–torsion and higher‐dimensional X the question remains open in general (and indeed one expects counter–examples to appear).

— If one adds extra hypotheses (for instance that X admits a semistable or strictly semistable model at the bad primes, or if one knows a very strong form of p–adic (log)–purity or a bound on wild ramification à la Abbes–Saito, or if one assumes resolution of singularities and then uses de Jong’s alterations plus a careful study of vanishing cycles) then one can force R^1i^! ℤ/n to become a bona–fide constructible, finite–rank sheaf on each irreducible component of X_p, and from that one deduces H^0(R^1i^! ℤ/n) finite and hence H^2(X, ℤ/n) finite. But {\it in the absence of such strong hypotheses} one simply does not know how to control the wild part, and the general case is still open.