Limits are additive, but suprema and infima aren't?

You're comparing apples to oranges. Limits are taken over sequences, while supremum and infimum are defined for sets. This is where the concepts of $\lim\sup $ and $\lim\inf$ are most useful. But in general you won't have full additivity for supremum limits and infimum limits in sequences in a similar way to how interference in wave patterns can be either destructive or constructive. If you have the sequence: $$1,0,1,0,1,0,1,0,...$$ and the sequence $$0,2,0,2,0,2,0,2,...$$ Then their sum is: $$1,2,1,2,1,2,1,2,...$$

Because the "lows" of one coincide with the "highs" of the other, and if you notice the lim sup and lim inf are not additive.

There's a sense in which $\limsup$ is a limit. If $\{a_n\}$ is a sequence, then $$\limsup_{n\to\infty} a_n =\lim_{n\to\infty} \left(\sup_{m>n} a_m\right)$$

Now, limits are additive, but that just means that:

$$\limsup_{n\to\infty} a_n + \limsup_{n\to\infty} b_n = \lim_{n\to\infty} \left(\sup_{m>n} a_m + \sup_{m>n} b_m\right)$$

Unfortunately, $$\sup_{m>n} a_m + \sup_{m>n} b_m \neq \sup_{m>n} (a_m+b_m)$$

in general. We can only say:

$$\sup_{m>n} a_m + \sup_{m>n} b_m \geq \sup_{m>n} (a_m+b_m)$$