This is an interesting observation made by one of my friends. After thinking for a bit, I think I was able to find the solution which I will write as an answer. I feel this is worth posting, both because it's interesting and perhaps pedagogically useful. For simplicitly, we will work with $\limsup$ but $\liminf$ is basically the same thing.
For any non-empty, bounded $A,B \subset \mathbb{R}$, we define $P = \{x + y : x \in A, y \in B\}$. It's easy to see that $\sup P = \sup A + \sup B$. Therefore, \begin{align}\lim \sup a_n + \lim \sup b_n &= \lim_{m \to \infty} \sup_{n \geq m} a_n + \lim_{m \to \infty} \sup_{n \geq m}b_n = \lim_{m \to \infty} \left(\sup_{n \geq m} a_n + \sup_{n \geq m}b_n\right) \\[0.3cm]&= \lim_{m \to \infty} \left(\sup_{n \geq m} \left(a_n + b_n\right)\right) = \lim \sup \ (a_n + b_n)\end{align}
This is wrong, however. The relationship between the left-most expression and the right-most is $\geq$. Where is the mistake?