If a language is not Turing reducible to two languages, may it still be Turing reducible to their "union"?
A classical theorem of Sacks states that if $L$ is not computable, then it is almost surely not computable relative to a random oracle. In other words, if $O$ is a random oracle, then the probability that $L$ is computable given $O$ is zero.

Now take your favorite uncomputable $L$, and choose $L_1,L_2$ at random among all languages such that $L = L_1 \Delta L_2$ (here $\Delta$ is symmetric difference). Individually, $L_1,L_2$ are random oracles, and so almost surely $L$ is not computable given just $L_1$ or just $L_2$. However, it is clearly computable given both.

Another solution is to take any two incomparable Turing degrees $L_1,L_2$ and $L = \{ 0x : x \in L_1 \} \cup \{ 1x : x \in L_2 \}$. The proof then follows practically by definition.