Does there exist a sequence of jointly normal random variables $(X_n)_{n=1}^\infty$ with $$\mathbb{E} X_n = 0 \quad \text{and} \quad \mathbb{E} X_n^2 = 1$$ for all $n \ge 1$, for which $X_n \to 0$ almost surely?
Clearly if all the $X_n$ are independent this does not hold, but if for example $X_1 = X_2 = \dots$, then this fails only slightly: There does not exist any deterministic constant $C > 0$ such that almost surely $|X_k| < C$ for large enough $k$.
Convergence in distribution gives $${\cal N}(0,1),\, {\cal N}(0,1),\, {\cal N}(0,1),\, {\cal N}(0,1),\cdots\Longrightarrow {\cal L}(X),$$ where ${\cal L}(X)$ is the distribution of $X$. The only possible limit point of a constant sequence is the constant itself, so ${\cal L}(X)={\cal N}(0,1).$