Determining if a random variable is lognormal I'm struggling to understand a part of the solution to this question: If $X$ and $Y$ are lognormal random variables, is their product $XY$ lognormally distributed?

The solution I have suggests looking at $ln(XY) = ln(X) + ln(Y)$. However, it says that $XY$ being lognormal depends on if $ln(X)$ and $ln(Y)$ are joint Normal. That is, $ln(X) + ln(Y)$ may not be normally distributed even if they are individually normally distributed implying $XY$ will not be lognormal. 

But I don't understand how $ln(X) + ln(Y)$ could be possibly non-Normal if each of them are Normal, as the sum of two normals is a normal.
The sum of two joint normal distributed random variables is again normal.

It is important that the random variables are actually joint normal, as the assertion can fail otherwise. The classical example is the following:

Let $X$ and $Z$ be independent random variables, where $X$ is standard Gaussian and $Z$ is Rademacher distributed, i.e. $P(Z = 1) = P(Z = -1) = \frac{1}{2}$. 

Then the random variables $X$ and $Y = ZX$ are both normal [and even uncorrelated!], but their sum isn't normal since $P(X + Y = 0) = P(X(Z + 1) = 0) = P(Z=-1)=\frac{1}{2}$.
As a counter-example to your statement that the "sum of two normals is a normal":

Suppose $W \sim N(0,1)$, $S=1$ when $|W| \lt 1$, and $S=-1$ when $|W| \ge 1$. Then $V=SW $ is also normally distributed as $N(0,1)$ but $|W+V| \lt 2$ so in this case the sum $W+V$ is not normally distributed.