Say I observe n univariate random variables $X_1, \dots, X_n$ that are each $N(\mu, \sigma^2)$ with common correlation $\rho$. Is it possible that these are jointly normal? If so, what are the conditions and how would I know if they are jointly normal.
There are no conditions based only on the marginal pdfs that can ensure joint normality.
Let $\phi(\cdot)$ denote the standard normal density. Then, if $X$ and $Y$ have joint pdf
$$f_{X,Y}(x,y) = \begin{cases} 2\phi(x)\phi(y), & x \geq 0, y \geq 0,\\
2\phi(x)\phi(y), & x < 0, y < 0,\\
0, &\text{otherwise},\end{cases}$$
then $X$ and $Y$ are (positively) correlated standard normal random variables (work out the marginal densities to verify this if it is not immediately obvious) that do not have a bivariate joint normal density. So, given only that $X$ and $Y$ are correlated standard normal random variables, how can we tell whether $X$ and $Y$ have the joint pdf shown above or the bivariate joint normal density with the same correlation coefficient ?
In the opposite direction, if $X$ and $Y$ are independent random variables (note the utter lack of mention of normality of $X$ and $Y$) and $X+Y$ is normal, then $X$ and $Y$ are normal random variables (Feller, Chapter XV.8, Theorem 1).