This question bothers me as I am not sure if we could drop the assumption of uncountable cofinality:
Let $\kappa$ be an uncountable cardinal and let $x_\alpha, y_\alpha$ be collections of real numbers indexed by ordinals $\alpha<\kappa$. Suppose that
$$1\leqslant |x_\alpha + y_\alpha|$$
for all $\alpha$. Does it follow that there exists $\theta\in (0,1)$ such that at least one of the sets
$$\{\alpha\colon |x_\alpha| > \theta\}\text{ or }\{\alpha\colon |y_\alpha| > \theta\}$$ has cardinality $\kappa$?
This is trivial if $\operatorname{cf}(\kappa)\geqslant \omega_1$.
Fix your favorite positive $\theta<\frac12$. I claim that your conclusion is true for this choice of $\theta$. To prove it, suppose not. So the number of $\alpha$'s with $|x_\alpha|>\theta$ is some cardinal $\lambda<\kappa$, and the number of $\alpha$'s with $|y_\alpha|>\theta$ is some cardinal $\mu<\kappa$. Since $\lambda+\mu=\max\{\lambda,\mu\}<\kappa$, there is an $\alpha<\kappa$ such that both $|x_\alpha|$ and $|y_\alpha|$ are $<\theta$. But then $|x_\alpha+y_\alpha|<2\theta<1$, contrary to hypothesis.