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Theorem (Artin 1971 [1]): Let $X$ be a quasi-compact AF-scheme. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for all $i$.

Theorem (Schröer 2003 [7]): Let $X$ be a Noetherian scheme and let $n$ be an integer such that $n \le a(X)$. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for $i \le n$ and an injection for $i = n+1$.

Example: For an example of a scheme $X$ and an abelian sheaf $\mathscr{F}$ on the etale site of $X$ for which the map $$ \check{\mathrm{H}}^{2}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{2}(X,\mathscr{F}) $$ is not an isomorphism, see the answers to this question. The current answers discuss the cases (1) $\mathscr{F}$ is a constant sheaf and (2) $\mathscr{F} = \mathcal{O}_{X}$.

(Gabber) For an example of a scheme $X$ for which the map $$ \check{\mathrm{H}}^{2}(X,\mathbb{G}_{m}) \to \mathrm{H}_{\mathrm{et}}^{2}(X,\mathbb{G}_{m}) $$ is not an isomorphism, let $R$ be a normal noetherian strictly henselian local ring of dimension $\ge 2$ whose punctured spectrum $U$ has nonzero Picard group (see e.g. this and this for examples of such $R$), and let $X$ be the gluing of two copies of $\operatorname{Spec} R$ along $U$. This is a local version of the counterexample to $\operatorname{Br} = \operatorname{Br}'$ due to Edidin, Hassett, Kresch, Vistoli [9].

Theorem (Artin 1971 [1]): Let $X$ be a quasi-compact AF-scheme. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}{}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for all $i$.

Theorem (Schröer 2003 [7]): Let $X$ be a Noetherian scheme and let $n$ be an integer such that $n \le a(X)$. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}{}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for $i \le n$ and an injection for $i = n+1$.

Example: For an example of a scheme $X$ and an abelian sheaf $\mathscr{F}$ on the etale site of $X$ for which the map $$ \check{\mathrm{H}}{}^{2}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{2}(X,\mathscr{F}) $$ is not an isomorphism, see the answers to this question. The current answers discuss the cases (1) $\mathscr{F}$ is a constant sheaf and (2) $\mathscr{F} = \mathcal{O}_{X}$.

(Gabber) For an example of a scheme $X$ for which the map $$ \check{\mathrm{H}}{}^{2}(X,\mathbb{G}_{m}) \to \mathrm{H}_{\mathrm{et}}^{2}(X,\mathbb{G}_{m}) $$ is not an isomorphism, let $R$ be a normal noetherian strictly henselian local ring of dimension $\ge 2$ whose punctured spectrum $U$ has nonzero Picard group (see e.g. this and this for examples of such $R$), and let $X$ be the gluing of two copies of $\operatorname{Spec} R$ along $U$. This is a local version of the counterexample to $\operatorname{Br} = \operatorname{Br}'$ due to Edidin, Hassett, Kresch, Vistoli [9].

Later Schroër refined Artin's result as follows:

Theorem (Schroër 2003 [7]): Let $X$ be a Noetherian scheme and let $n$ be an integer such that $n \le a(X)$. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for $i \le n$ and an injection for $i = n+1$.

[7] S. Schroër, "The bigger Brauer group is really big", Journal of Algebra 262 (2003) pp 210–225, doi:10.1016/S0021-8693(03)00026-7, arXiv:math/0108135.

Later Schröer refined Artin's result as follows:

Theorem (Schröer 2003 [7]): Let $X$ be a Noetherian scheme and let $n$ be an integer such that $n \le a(X)$. Then for any abelian sheaf $\mathscr{F}$ on the etale site of $X$, the map $$ \check{\mathrm{H}}^{i}(X,\mathscr{F}) \to \mathrm{H}_{\mathrm{et}}^{i}(X,\mathscr{F}) $$ is an isomorphism for $i \le n$ and an injection for $i = n+1$.

[7] S. Schröer, "The bigger Brauer group is really big", Journal of Algebra 262 (2003) pp 210–225, doi:10.1016/S0021-8693(03)00026-7, arXiv:math/0108135.

[1] Artin, "On the joins of Hensel rings", (1971)

[2] Benoist, "Quasi-projectivity of normal varieties", International Mathematics Research Notices, vol 2013, no 17 (2012)

[3] Farnik, "On strengthening of the Kleiman-Chevalley criterion", Proceedings of the AMS, vol 141, no 11 (2013), link

[4] Gross, "Tensor generators on schemes and stacks", link

[5] Kleiman, "Toward a numerical theory of ampleness", Annals of Mathematics (1966)

[6] Milne, Etale Cohomology (1980)

[7] Schroër, "The bigger Brauer group is really big", Journal of Algebra, vol. 262 (2003)

[8] Stacks Project

[9] Edidin, Hassett, Kresch, Vistoli, "Brauer groups and quotient stacks", American Journal of Mathematics, Vol. 123, No. 4 (2001)

[1] M. Artin, "On the joins of Hensel rings", Advances in Mathematics 7 (1971) pp 282–296, doi:10.1016/S0001-8708(71)80007-5, core.ac.uk.

[2] O. Benoist, "Quasi-projectivity of normal varieties", International Mathematics Research Notices, vol 2013, no 17 (2012) pp 3878–3885, doi:10.1093/imrn/rns163, arXiv:1112.0975.

[3] M. Farnik, "On strengthening of the Kleiman-Chevalley criterion", Proceedings of the AMS 141 no 11 (2013) pp 4005-4013, doi:10.1090/S0002-9939-2013-11695-3.

[4] P. Gross, "Tensor generators on schemes and stacks", Algebraic Geometry 4 (4) (2017) pp 501–522, doi:10.14231/ag-2017-026, arXiv:1306.5418.

[5] S. Kleiman, "Toward a numerical theory of ampleness", Annals of Mathematics 84 No. 3 (1966) pp 293–344, doi:10.2307/1970447.

[6] J.S. Milne, Etale Cohomology, Princeton University Press (1980) JSTOR (subscription needed).

[7] S. Schroër, "The bigger Brauer group is really big", Journal of Algebra 262 (2003) pp 210–225, doi:10.1016/S0021-8693(03)00026-7, arXiv:math/0108135.

[8] Stacks Project, https://stacks.math.columbia.edu/.

[9] Edidin, Hassett, Kresch, Vistoli, "Brauer groups and quotient stacks", American Journal of Mathematics 123 No. 4 (2001), doi:10.1353/ajm.2001.0024, JSTOR, arXiv:math/9905049.